Worst-Case Examples for the Computation of Persistent Homology

W ORST-CASE EXAMPLES F OR THE COMPUT A TION OF PERSISTENT HOMOLOGY UZA Y ÇETIN AND ER GÜN Y ALÇIN Abstra ct. W e construct worst-case examples for the standard reduction algorithm for computing p ersisten t homology . Our constructions are similar to the w orst-case examples in tro duced by Morozo v, but w e replace the single-triangle arrangemen t with a strip of base and fin triangles. This structure allo ws us to give an explicit algorithm for their construction and to p erform exp eriments comparing the runtime of different v ersions of the reduction algorithm. W e further show that, after suitable edge and triangle subdivisions, these strip examples remain worst-case and can b e realized as clique complexes of filtered graphs, and hence as Vietoris–Rips complexes of finite point clouds for a sequence of scale parameters. 1. Intr oduction Due to the p opulation increase and adv ances in in ternet and digital tec hnologies, data has b ecome one of the most imp ortant asp ects of our liv es. While classical mathematical metho ds such as statistical and clustering techniques are useful in data analysis, T op ological Data Analysis (TD A) has emerged as an imp ortant new tool for analyzing data in last 20 y ears [2]. TD A is curren tly used in many areas, including genomics, rob otics, medical imaging, cancer research, the spread of infectious diseases, financial netw orks, and natural language analysis (see, for example, [3], [6], [13], and [5]). T op ological Data Analysis uses metho ds from algebraic top ology to analyze data. One of the main methods in TDA is p ersistent homology , which was dev eloped by Edelsbruner, Letsc her, Zomorodin, Carlsson, and others (see [8], [15]). In most of the applications, TD A via p ersisten t homology takes as input a finite metric space embedded in R m and pro duces p ersisten t diagrams as output. One of the reasons that makes the p ersistent homology so effectiv e is that it can be computed b y a simple algorithm called the reduction algorithm. There are many computer pack ages based on different versions of the reduction algorithm that computes the p ersisten t diagram of a filtered simplicial complex (see [12] for a comparison of the run time p erformance of different pack ages). In [10], Morozo v constructed a family of filtered simplicial complexes that pro vide worst- case examples for the reduction algorithm, exhibiting a runtime of Ω( N 3 ) , where N denotes the num ber of simplices. The main purp ose of this pap er is to give a new family of worst-case examples similar to Morozo v’s examples, but they are constructed b y successively adding base and fin triangles to a strip of triangles instead of adding triangles inside a single triangle. W e refer to these new set of w orst-case examples as strip examples . Because of their strip structure, it is easier to giv e an explicit algorithm for their construction and to p erform exp erimen ts comparing the runtime of different versions of the reduction algorithm. In Theorem 3.2, w e show that the run time of the reduction algorithm on the strip worst-case examples is again Ω( N 3 ) , where N is the n um b er of simplices. In Appendix A, we present Date : March 18, 2026. 2020 Mathematics Subje ct Classific ation. 55N31 (primary), 68W40, 68R10 (secondary). Key wor ds and phr ases. T op ological Data Analysis, Persisten t Homology , Reduction Algorithm, Algorith- mic Complexit y , Clique Complexes. 1 2 UZA Y ÇETIN AND ERGÜN Y ALÇIN a simple pseudo-co de for constructing the strip examples for an y num ber of triangles. In Section 4 w e use this co de to p erform computational exp eriments and rep ort on the results comparing the runtimes of different versions of the reduction algorithm suc h as the standard reduction algorithm [8], the reduction algorithm with a twist [4], and the Lo ok-ahead v arian t of the reduction algorithm [11]. In Section 5, we sho w that after mo difying the strip examples by sub dividing some of the triangles and edges, they can b e realized as clique complexes of filtered graphs. Then we sho w that these filtered complexes can be realized as filtered graphs asso ciated to a finite data cloud in some Euclidean space. As a consequence w e obtain that these mo dified strip examples can b e realized as Vietoris–Rips complexes of finite p oint clouds for some sequence of real n umbers. 2. Ma trix Reduction Algorithms W e refer the reader to [6], [13], and [5] for basic definitions on persistent homology . Here w e only giv e the definitions related to filtered simplicial complexes that will be used in the pap er to explain the reduction algorithm. Definition 2.1. A simplicial c omplex K consists of a pair ( V , S ) where V is the set of v ertices of K , and S is a set of subsets of V , called the simplices of K , satisfying the following prop erties: (1) for all v ∈ V , { v } ∈ S , (2) if σ ∈ S and τ ⊆ σ , then τ ∈ S . A simplex σ in K is called a k -simplex or a k -dimensional simplex if | σ | = k + 1 . F or a simplex σ in K , a subset τ ⊆ σ is called a fac e of σ . If dim τ = dim σ − 1 , then τ is called a fac et of σ . W e say the simplicial complex K is finite if its v ertex set V is finite. Throughout the pap er we assume all the simplicial complexes are finite. The dimension of K is the largest d such that K has a d -dimensional simplex. W e say that the simplicial complex K ′ = ( V ′ , S ′ ) is a sub complex of K = ( V , S ) if V ′ ⊆ V and S ′ ⊆ S . In this case we write K ′ ⊆ K . Definition 2.2. A nested sequence of simplical complexes ∅ ⊆ K 1 ⊆ K 2 ⊆ · · · ⊆ K m = K is called a filtr ation of K . A simplicial complex with a filtration is called a filter e d simplicial c omplex . A filtration function on a simplicial complex K = ( V , S ) is a function f : S → R satisfying f ( τ ) ≤ f ( σ ) whenever τ ⊆ σ . F or eac h filtered complex K , w e can define a filtration function f : K → Z + = { 1 , 2 , . . . } such that for each simplex σ , f ( σ ) is the smallest i ∈ Z such that σ ∈ K i . Conv ersely , given a filtration function f on a complex K , w e can c hoose finitely man y real num bers r 1 , . . . , r m and define the filtration on K b y taking K i ⊆ K to b e the sub complex consists of simplices σ ∈ K suc h that f ( σ ) ≤ r i . Let K = ( V , S ) b e a filtered simplicial complex ∅ ⊆ K 1 ⊆ K 2 ⊆ · · · ⊆ K m = K with asso ciated filtration function f : K → Z + . Consider the chain complex C ∗ ( K ) of K in F 2 - co efficien ts. W e can consider the chain complex C ∗ ( K ) as a graded F 2 -v ector space C ∗ ( K ) = L d p =0 C p ( K ) , where d = dim( K ) , and consider the family of boundary maps ∂ ∗ = { ∂ p : C p ( K ) → C p − 1 ( K ) } d p =1 as a linear transformation of degree -1 b et w een graded v ector spaces. T o define a matrix representing the b oundary map ∂ ∗ : C ∗ ( K ) → C ∗ ( K ) , we assume that the simplices in K are totally ordered, i.e. sorted as a list σ 1 , . . . , σ N , in such a wa y that it satisfies the follo wing prop erties: WORST-CASE EXAMPLES FOR THE COMPUT A TION OF PERSISTENT HOMOLOGY 3 (1) If σ i is a face of σ j , then i < j . (2) If f ( σ i ) < f ( σ j ) , then i < j . Suc h an ordering of simplices of K is called or dering c omp atible with the filtration. Because of condition (1), the b oundary matrix is strictly upp er triangular. Definition 2.3. Let K b e a filtered simplicial complex and S = { σ 1 , σ 2 , . . . , σ N } b e the set of simplices of K ordered with an ordering compatible with the filtration. The b oundary matrix of K is the N × N -matrix D with entries in F 2 suc h that D i,j = ( 1 if σ i is a facet of σ j 0 otherwise. F or each j , the j -th column of the b oundary matrix D is denoted by D j . 2.1. The Standard Algorithm for Reduction. It is shown in [8] that the computation of the p ersistent diagram can b e accomplished by means of a simple algorithm that brings the boundary matrix in to a reduced form. T o define this algorithm w e need to introduce a function called the low function. Definition 2.4. F or ev ery nonzero v ector v with en tries in F 2 , the low of v is defined by lo w( v ) = max( { i | v [ i ] = 1 } ) where v [ i ] denotes the i -th entry of v . W e set low( 0 ) = 0 . The standard reduction algorithm is similar to the Gaussian elimination but it is p erformed on the columns. F or each j running from 1 to N , the j -th column D j is mo dified as follows: If low( D i ) = low( D j ) for some i < j , then the vector D j is replaced b y D i + D j . Here the addition is performed in the field F 2 . Pseudoco de for standard algorithm can b e giv en in Algorithm 1. Algorithm 1 Standard Reduction Algorithm Input: Boundary Matrix D = [ D 1 | D 2 | · · · | D N ] ∈ F N × N 2 Output: Reduced Matrix R ∈ F N × N 2 R ← D ; for j = 1 to N do ▷ Ω( N ) while ∃ i < j with low( D i ) = lo w( D j )  = 0 do ▷ Ω( N ) D j ← D i + D j ▷ Ω( N ) end while end for F or ev ery k = 1 , . . . , d , the persistent diagram dgm k ( K ) for a filtered complex K of di- mension d is the set of all p ersisten t pairs ( f ( σ i ) , f ( σ j )) such that f ( σ i ) is the filtration level where a cohomology class at dimension k is b orn, and f ( σ j ) is the lev el where it dies (see [6] for details). Note that if R is the reduced matrix obtained from the boundary matrix of K after applying the reduction algorithm, then the pair ( f ( σ i ) , f ( σ j )) is in the p ersistent diagram dgm k ( K ) if and only if σ i is a k -dimensional simplex and i = low( R j )  = 0 . F or all the columns of R with R i = 0 , we can conclude that there is a birth at f ( σ i ) . If R i = 0 for some i and there is no low at R i,j for an y j , then we sa y there is a p ersistence pair ( f ( σ i ) , ∞ ) . 4 UZA Y ÇETIN AND ERGÜN Y ALÇIN Example 2.5. Let K b e the filtered simplicial complex whose simplices in the order of app earance are given as follows: K = { [1] , [2] , [3] , [1 , 2] , [1 , 3] , [2 , 3] , [1 , 2 , 3] } . The b oundary matrix of K is: D =           [1] [2] [3] [1 , 2] [1 , 3] [2 , 3] [1 , 2 , 3] [1] 0 0 0 1 1 0 0 [2] 0 0 0 1 0 1 0 [3] 0 0 0 0 1 1 0 [1 , 2] 0 0 0 0 0 0 1 [1 , 3] 0 0 0 0 0 0 1 [2 , 3] 0 0 0 0 0 0 1 [1 , 2 , 3] 0 0 0 0 0 0 0           Then, the standard reduction algorithm is applied to the b oundary matrix D as follows: D =           0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0           C 5+ C 6 − → C 6 − − − − − − − − →           0 0 0 1 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0           C 4+ C 6 − → C 6 − − − − − − − − →           0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0           = R If w e tak e the filtration function to b e the function that assigns σ i to the index i , then from the indices of the lo ws in the reduction matrix, w e can conclude that the set of persistent pairs for the filtered simplicial complex K is dgm 0 ( K ) = { (1 , ∞ ) , (2 , 4) , (3 , 5) } and dgm 1 ( K ) = { (6 , 7) } . 2.2. Reduction Algorithm with a T wist. Chen and Kerb er [4] introduced a v ariation of the reduction algorithm called the reduction algorithm with a twist. The idea b ehind this new algorithm is that if we do the reduction b y first applying it to higher dimensional simplices, w e will find a p ersistent pair ( f ( σ i ) , f ( σ j )) after reducing the j -column b efore pro cessing the i -th column. Since this pair tells us that the simplex σ i creates a homology class, we must ha v e the i -th column in the reduced matrix equal to zero. So w e can clear this column, i.e. set to zero without doing any column additions. This sa v es many unnecessary column op erations and mak es the reduction algorithm run muc h faster. T o in tro duce the pseudoco de for this algorithm, first note that the b oundary matrix can b e decomp osed to smaller b oundary matrices for linear maps ∂ p : C p ( K ) → C p − 1 ( K ) for p = 1 , . . . , d . The reduction algorithm can be describ ed using these smaller b oundary matrices. Pseudo co de for the reduction algorithm with a twist is given in Algorithm 2. Example 2.6. Let D denote the b oundary matrix of the filtered simplicial complex as in Example 2.5. Then, the reduced matrix R is obtained using Chen and Kerber’s algorithm after applying only one step: D =           0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0           clear the 6 -th column − − − − − − − − − − − − − →           0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0           = R WORST-CASE EXAMPLES FOR THE COMPUT A TION OF PERSISTENT HOMOLOGY 5 Algorithm 2 Reduction Algorithm with a T wist Input: Boundary Matrix D = [ D 1 | D 2 | · · · | D N ] ∈ F N × N 2 Output: Reduced Matrix R ∈ F N × N 2 R ← D ; L : [0 , 0 , . . . , 0] for δ = d, . . . , 1 do for j = N , . . . , 1 do if dim σ j = δ then while R j  = 0 and L [lo w( R j )]  = 0 do R j ← R j + R L [low( R j )] end while if R j  = 0 then i ← low( R j ) L [ i ] ← j R i ← 0 end if end if end for end for The twist algorithm is a v ery efficient algorithm, and it is used in all the current p ersistent homology calculation pac k ages suc h as Gudhi, Phat, and Ripser. W e discuss the runtime of the t wist algorithm in Section 2.4. 2.3. Reduction Algorithm with Lo ok-ahead. In a recen t pap er, Morozov and Skraba [11] introduced an algorithm for p ersistent homology in matrix m ultiplication time. In this pap er they also discuss a lo ok-ahead v ariant of the standard reduction algorithm. The main idea of this algorithm is that if the low of the column R j is i  = 0 , then for every column R j ′ with j ′ > j and R i,j ′ = 1 , w e add column R j to R j ′ to mak e R i,j ′ = 0 . Pseudo co de for this algorithm is giv en in Algorithm 3. Algorithm 3 Reduction Algorithm with Lo ok-ahead Input: Boundary Matrix D = [ D 1 | D 2 | · · · | D N ] ∈ F N × N 2 Output: Reduced Matrix R ∈ F N × N 2 R ← D ; for j = 1 to N do ▷ Ω( N ) if lo w( R j )  = 0 then i ← low( R j ) for j ′ > j and R [ i, j ′ ] = 1 do ▷ Ω( N ) R [ · , j ′ ] ← R [ · , j ′ ] + R [ · , j ] ▷ Ω( N ) end for end if end for Example 2.7. Let D denote the b oundary matrix as in Example 2.5. Then, with the lo ok- ahead reduction algorithm, the reduced matrix R is obtained as follows: 6 UZA Y ÇETIN AND ERGÜN Y ALÇIN D =           0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0           C 4+ C 6 − → C 6 − − − − − − − − →           0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0           C 5+ C 6 − → C 6 − − − − − − − − →           0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0           = R Note that in this example the reduced matrix is obtained using exactly the same num b er of op erations as in the standard algorithm. This is not true in general. There are examples where the lo ok-ahead reduction algorithm p erforms b etter than the reduction algorithm. W e discuss the runtime of the lo ok-ahead v arian t of the reduction algorithm applied to the strip w orst-case examples in Section 4. 2.4. The runtime of reduction algorithms. It is shown that for all three v arian ts of the reduction algorithm, the runtime is Ω( N 3 ) in the worst-case where N is the num b er of simplices. F or the standard reduction algorithm, w e see this by calculating run time of the differen t steps of the algorithm. In this algorithm we ha v e a for lo op with N elements so it runs in Ω( N ) time. The while lo op also runs in Ω( N ) . Inside these lo ops the summation of t w o vectors of size N is p erformed in Ω( N ) time. So the runtime of the standard reduction algorithm is Ω( N 3 ) . The runtime calculation for the reduction algorithm with a t wist is similar. In this algo- rithm, the first t wo for lo ops run in Ω( N ) , the while lo op runs in Ω( N ) , and the additions are p erformed in Ω( N ) time. So the complexity of the reduction algorithm with a t wist is also Ω( N 3 ) . It is shown that in practice the twist algorithm is muc h more efficient b ecause the b oundary matrix is a sparse matrix in general. There is an explicit calculation for the runtime of this algorithm in terms of the num b er of p ersisten t pairs in the reduced matrix (see [4]). In the exp erimen ts that we performed with random data p oints of size 10, we observ ed that the n um b er of v ector additions that are p erformed go es down drastically b y using the reduction algorithm with t wist. Our exp eriments with the worst case strip examples also supp ort this claim (see Section 4). In the lo ok-ahead v arian t of the standard algorithm, w e ha v e tw o for lo ops each ha ving run time Ω( N ) , and there is a vector addition with runtime Ω( N ) . Hence the lo ok-ahead algorithm also runs in Ω( N 3 ) . Similar to the twist algorithm we can p erform some calculations to compare the efficiency of the lo ok-ahead reduction algorithm with other algorithms. In our exp erimen ts with a random data set of size 10, we obtained that the run time of the lo ok-ahead reduction algorithm is b etter than the runtime of the standard reduction algorithm. 3. An Al terna tive Worst-case Example: Strip In this section w e construct a worst-case example for the reduction algorithm, called the strip example, similar to the worst-case examples introduced by Morozov in [10]. F or each p ositive in teger n , the strip worst-c ase c omplex X ( n ) is a filtered simplicial com- plex consisting of n base triangles sitting on a plane and n fin triangles with one edge lying on the plane and one vertex lying outside the plane. W e place these base triangles and fin trian- gles as sho wn in Figure 1. The base triangles are lab eled i and the fin triangles are lab eled -i for i = 1 , . . . , n . The vertices of these triangles are lab eled v 1 , . . . , v n +2 and f 1 , . . . , f n . The lab eling of the base triangles starts from the center and then alternates left and right. WORST-CASE EXAMPLES FOR THE COMPUT A TION OF PERSISTENT HOMOLOGY 7 W e lab el the horizontal edges of the base triangles 1 ′ , . . . , n ′ suc h that i ′ is the horizon tal edge of the base triangle i . The vertical base edges are lab eled 1 , . . . , n + 1 so that 1 is the edge b etw een v 1 and v 2 , and for all i = 2 , . . . , n + 1 , the edge i is b et ween v i − 1 and v i +1 as sho wn in Figure 1. The base edge with lab el n + 1 is also lab eled as 0 ′ to make it easier to justify its place in the filtration. The edges of the fin triangles are lab eled − 1 , . . . , − n and − 1 ∗ , . . . , − n ∗ . The edge − 1 is b et ween the vertices f 1 and v 2 , and the edge − 1 ∗ is b etw een f 1 and v 1 . F or all i = 2 , . . . , n , the edge − i is b et w een the vertices f i and v i +1 , and the edge − i ∗ is b et w een f i and v i − 1 . Note that there is one fin triangle ab ov e each vertical base edge except the edge with label n + 1 = 0 ′ . n-1 n-3 1 2 3 4 5 n n-2 . . . . . . . . . . . . 3 ′ 2 ′ ( n − 2) ′ ( n − 3) ′ 1 ′ 4 ′ 5 ′ n ′ ( n − 1) ′ n + 1 = 0 ′ 1 2 3 4 5 6 n n − 1 − n − n ∗ -n f n − 1 − 1 ∗ -1 f 1 ( − n + 1) ∗ − n + 1 -n+1 f n − 1 v n +1 v n − 5 v n − 3 v n − 1 v 6 v 4 v 2 v 1 v 3 v 5 v 7 v n − 2 v n − 4 v n v n +2 Figure 1. The simplicial complex X ( n ) for n ≡ 1 (mo d 4) . The strip worst-case examples grow inductiv ely by adding triangles. T o reach from X ( n ) to X ( n + 1) we just add a fin triangle ab ov e the edge n + 1 and a base triangle next to the edge with lab el n . Note that the simplicial complex X ( n ) has 2 n + 2 vertices, 4 n + 1 edges, and 2 n triangles. The num b er of simplices of X ( n ) is N = 8 n + 3 . The Euler characteristic of X ( n ) is equal to 1 since it has homology isomorphic to the homology of a p oint. T o define the filtration on X ( n ) , w e order the simplices of the complex using the following ordering: (1) F in in v ertices f 1 , . . . , f n , (2) Base vertices v 1 , . . . , v n +2 , (3) Horizon tal base edges that destroy comp onents n ′ , ( n − 1) ′ , . . . , 1 ′ , (4) B ase edge n + 1 which is also lab eled as 0 ′ , (5) F in edges that destroy fin comp onents − n ∗ , . . . , − 1 ∗ , (6) F in edges that creates cycles − n, − n + 1 , . . . , − 1 , (7) V ertical base edges n, n − 1 , . . . , 1 , (8) B ase triangles 1 , 2 , . . . , n , and (9) F in triangles -1 , -2 , . . . , -n . T o simplify calculations, in the pro of of Theorem 3.2 we apply the reduction algorithm only on the b oundary matrix for ∂ 2 : C 2 ( X ( n )) → C 1 ( X ( n )) . W e denote this matrix by B ( n ) and the asso ciated reduced matrix by R ( n ) . 8 UZA Y ÇETIN AND ERGÜN Y ALÇIN Example 3.1. F or n = 2 , the b oundary matrix B (2) and the reduced matrix R (2) are as follo ws: B (2) =               1 2 -1 -2 2 ′ 0 1 0 0 1 ′ 1 0 0 0 0 ′ 0 1 0 0 − 2 ∗ 0 0 0 1 − 1 ∗ 0 0 1 0 − 2 0 0 0 1 − 1 0 0 1 0 2 1 0 0 1 1 1 1 1 0               C 1 + C 2 → C 2 C 1 + C 3 → C 3 C 2 + C 3 → C 3 C 2 + C 4 → C 4 − − − − − − − →               1 2 -1 -2 0 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 0 0               = R (2) The pro cessing of the columns of the matrix B (2) can b e explained using the sparse matrix notation: 1 → (1 , 2 , 1 ′ ) Note that the low of the column for 1 is the row corresp onding to the edge 1 . In this case w e say the triangle 1 is paired with the edge 1 . F or the triangle 2 , we hav e 2 → (1 , 0 ′ , 2 ′ ) 1 + 2 → (1 , 2 , 1 ′ ) + (1 , 0 ′ , 2 ′ ) = (2 , 0 ′ , 1 ′ , 2 ′ ) This pairs the triangle 2 with the edge 2 . F or the fin triangles we hav e a similar pairing pro cess. -1 → (1 , − 1 , − 1 ∗ ) -1 + 1 → (1 , − 1 , − 1 ∗ ) + (1 , 2 , 1 ′ ) = (2 , − 1 , − 1 ∗ , 1 ′ ) -1 + 1 + ( 1 + 2 ) → (2 , − 1 , − 1 ∗ , 1 ′ ) + (2 , 0 ′ , 1 ′ , 2 ′ ) = ( − 1 , − 1 ∗ , 0 ′ , 2 ′ ) This mak es the fin triangle -1 pair with the edge − 1 . Finally for the fin triangle -2 , w e ha v e -2 → (2 , − 2 , − 2 ∗ ) -2 + ( 1 + 2 ) → (2 , − 2 , − 2 ∗ ) + (2 , 0 ′ , 1 ′ , 2 ′ ) = ( − 2 , − 2 ∗ , 0 ′ , 1 ′ , 2 ′ ) This pairs the triangle -2 with the edge − 2 . So the p ersistent pairs for the 1-dimensional homology H 1 are dgm 1 ( X (2)) = { (1 , 1 ) , (2 , 2 ) , ( − 1 , -1 ) , ( − 2 , -2 ) } . Here we listed the p ersistent pairs using the simplices since in the ab o v e matrices w e used the simplices for lab eling the columns and rows. Note that these pairs corresp ond to the indices of the lo ws in the reduced matrix R (2) . No w we pro ve our main result in this section. Theorem 3.2. L et B ( n ) denote the matrix for the b oundary map ∂ 2 : C 2 ( X ( n )) → C 1 ( X ( n )) . The runtime on the standar d r e duction algorithm applie d to B ( n ) has c omplexity Ω( n 3 ) . As a c onse quenc e, the c omplexity of the standar d r e duction algorithm for the b oundary matrix of X ( n ) is Ω( N 3 ) wher e N = 8 n + 3 is the numb er of simplic es of X ( n ) . WORST-CASE EXAMPLES FOR THE COMPUT A TION OF PERSISTENT HOMOLOGY 9 Pr o of. Extending the pairing algorithm given ab ov e for n = 2 , we see that the base triangles i will b e paired b y the base edges i with the following list of representativ es: 1 → (1 , 2 , 1 ′ ) 1 + 2 → (1 , 2 , 1 ′ ) + (1 , 3 , 2 ′ ) = (2 , 3 , 1 ′ , 2 ′ ) 1 + 2 + 3 → (2 , 3 , 1 ′ , 2 ′ ) + (2 , 4 , 3 ′ ) = (3 , 4 , 1 ′ , 2 ′ , 3 ′ ) . . . 1 + 2 + · · · + n-1 → ( n − 1 , n, 1 ′ , 2 ′ , · · · , ( n − 1) ′ ) 1 + 2 + · · · + n → ( n, 0 ′ , 1 ′ , 2 ′ , · · · , n ′ ) This table should b e read in the following w a y . The b oundary of 1 is (1 , 2 , 1 ′ ) as an ordered 3-tuple of edges. So 1 is paired with 1 . The b oundary of the triangle 2 is (1 , 3 , 2 ′ ) . Since 1 is already paired with 1 , we need to add the column of 1 to the column of 2 . This creates a c hain 1 + 2 with b oundary (2 , 3 , 1 ′ , 2 ′ ) . So the triangle 2 is paired with 2 . W e con tin ue this wa y until for all i , the triangle i is paired with edge i . F rom the table we see that in the reduced matrix R ( n ) , the column for i has i + 2 nonzero entries. F or each i , let us denote the chain 1 + · · · + i = ( i, i + 1 , 1 ′ , 2 ′ , . . . , i ′ ) b y c i . The complexit y of pro cessing the base triangles can b e calculated as follows: Since the complexit y of adding tw o vectors with a 1 and a 2 nonzero entries is equal to a 1 + a 2 , the complexit y of p erforming all the ab ov e additions is (3 + 3) + (4 + 3) + (5 + 3) + · · · + (( n + 1) + 3) = 1 2 ( n 2 + 9 n − 10) . No w we calculate the complexity of pro cessing the fin triangles. F or eac h i , the pro cessing of the fin triangle -1 is illustrated in the following table: -1 → (1 , − 1 , − 1 ∗ ) -1 + c 1 → (1 , − 1 , − 1 ∗ ) + (1 , 2 , 1 ′ ) = (2 , − 1 , − 1 ∗ , 1 ′ )  -1 + c 1  + c 2 → (2 , − 1 , − 1 ∗ , 1 ′ ) + (2 , 3 , 1 ′ , 2 ′ ) = (3 , − 1 , − 1 ∗ , 2 ′ )  -1 + c 1 + c 2  + c 3 → (3 , − 1 , − 1 ∗ , 2 ′ ) + (3 , 4 , 1 ′ , 2 ′ , 3 ′ ) = (4 , − 1 , − 1 ∗ , 1 ′ , 3 ′ )  -1 + c 1 + c 2 + c 3  + c 4 → (4 , − 1 , − 1 ∗ , 1 ′ , 3 ′ ) + (4 , 5 , 1 ′ , 2 ′ , 3 ′ , 4 ′ ) = (5 , − 1 , − 1 ∗ , 2 ′ , 4 ′ ) . . .  -1 + c 1 + c 2 + · · · + c n − 1  + c n → ( ( − 1 , − 1 ∗ , 0 ′ , 1 ′ , 3 ′ , 5 ′ , · · · , n ′ ) if n is o dd ( − 1 , − 1 ∗ , 0 ′ , 2 ′ , 4 ′ , 6 ′ , . . . , n ′ ) if n is even. This mak es the fin triangle -1 pair with the edge − 1 . There is a similar pro cess for eac h of the fin triangles. If we assume that the pro cessing of the fin triangle -1 is done by p erforming the additions -1 + c 1 + · · · + c n , then the total num ber of field additions on nonzero entries 10 UZA Y ÇETIN AND ERGÜN Y ALÇIN is equal to 3 + n X i =1 ( i + 2) = 1 2 ( n 2 + 5 n + 6) . In a similar wa y if w e pro cess the j -th fin triangle -j , w e see that we will need to p erform the addition -j + c j + · · · + c n , hence the total num b er of additions is 3 + n X i = j ( i + 2) = 1 2 ( n 2 + 5 n + − j 2 − 3 j + 10) . Summing these o ver all j = 1 , . . . , n , we obtain that the num ber of additions is equal to 1 6 n (2 n 2 + 9 n + 25) . A dding this to the complexit y of pro cessing the base triangles, we find the total num b er of additions is S ( n ) = 1 3 ( n 3 + 6 n 2 + 26 n − 15) . In particular, S ( n ) = Ω( n 3 ) . Since the num b er of simplices in the complex is N = 8 n + 3 , it follo ws that the worst-case running time of the standard reduction algorithm is Ω( N 3 ) . □ Remark 3.3. In our implemen tation, additions are carried out at eac h intermediate step, so the actual n umber of additions p erformed is usually larger than T ( n ) . 4. Experiments with the strip example One of our motiv ations for in tro ducing an alternative worst-case example is to b e able to write computer programs that use worst- case examples for p erformance tests. The linear structure of the strip example allo ws us to give a simple explicit algorithm for pro ducing the complex X ( n ) for ev ery n ≥ 1 . This algorithm can b e found in App endix A. The Python co de written using this algorithm and the results of the exp eriments that we run can b e found on the gith ub page: https://gith ub.com/Ergun1234/Morozo v_t yp e_Examples. After running the Python co de for constructing the strip example X ( n ) and running the standard reduction algorithm for the b oundary matrix of X ( n ) , we obtained the p ersistent diagrams for the strip examples. Figure 2 sho ws the filtered simplicial complexes X ( n ) for some v alues of n , and Figure 3 illustrates the corresp onding p ersistence diagrams. (a) n = 4 (b) n = 9 (c) n = 30 Figure 2. The 1 -skeletons of the strip worst-case simplicial complexes X ( n ) for differen t v alues of n . WORST-CASE EXAMPLES FOR THE COMPUT A TION OF PERSISTENT HOMOLOGY 11 (a) n = 4 (b) n = 9 (c) n = 30 Figure 3. Persistence diagrams for strip w orst-case examples X ( n ) for dif- feren t v alues of n . Blue p oin ts corresp ond to H 0 features and orange p oints corresp ond to H 1 features. The dashed line indicates the diagonal y = x . W e p erformed further exp eriments with the split worst-case examples and calculated the run time of the reduction algorithms, the standard reduction algorithm, the reduction algo- rithm with a twist, and the reduction algorithm with lo ok-ahead. Figure 4 shows the running times of these three algorithms where the runtime is calculated using a coun ter that tracks the n um b er of additions p erformed during the calculation of the reduced b oundary matrix. Fig- ure 5 sho ws the elapsed running times of the algorithms on a MacBo ok with an M5 pro cessor. (a) max n = 10 (b) max n = 40 Figure 4. Running times of the v arian ts of the reduction algorithm compared with a cubic function, for tw o v alues of max n (the maxim um v alue of n used in the exp eriments). These graphs show that the running time for the standard reduction algorithm and lo ok- ahead reduction algorithm are the same. The runtime of the reduction algorithm with twist is less than the other tw o algorithms. W e explain the reason for this in the Example 4.1. The elapse times for these three algorithms show that b oth twist and lo ok-ahead v arian ts of the reduction algorithm p erform muc h b etter than the standard algorithm when they are applied to the strip examples. 12 UZA Y ÇETIN AND ERGÜN Y ALÇIN (a) max n = 10 (b) max n = 40 Figure 5. Elapsed running time of different v arian ts of the reduction algo- rithm as a function of the parameter n , for tw o v alues of max n . Example 4.1. F or the strip example X (2) , consider the matrix B (1) for the b oundary map ∂ 1 : C 1 ( X (2)) → C 0 ( X (2)) . B (1) =         2 ′ 1 ′ 0 ′ − 2 ∗ − 1 ∗ − 2 − 1 2 1 f 1 0 0 0 0 1 0 1 0 0 f 2 0 0 0 1 0 1 0 0 0 v 1 1 0 0 1 1 0 0 1 1 v 2 0 1 1 0 0 0 1 0 1 v 3 0 1 0 0 0 1 0 1 0 v 4 1 0 1 0 0 0 0 0 0         T o reduce the first five columns of the b oundary matrix B (1) , w e p erform the column op erations C 1 + C 3 → C 3 and C 4 + C 5 → C 5 , requiring 8 field additions. Then we carry out 9 more column additions to clear the last four columns. These op erations require 36 additional field additions. If w e instead apply the reduction algorithm with twist, since we first pro cess the matrix B (2) , w e set the last four columns of B (1) to zero. This reduces the n umber of field additions from 44 to 8. Since the pro cessing of B (2) requires 27 field additions, the total n um b er of field additions p erformed for reducing the b oundary matrix decreases from 71 to 35 . This is the drop observed for n = 2 in the first graph in Figure 4. The following table sho ws the num ber of field additions p erformed by the v ariants of the reduction algorithm for n ≤ 5 . T able 1. Number of field additions performed b y the standard reduction algorithm and its v ariants. n Standard reduction With t wist Lo ok-ahead 1 22 10 22 2 71 35 71 3 145 85 145 4 247 155 247 5 380 248 380 WORST-CASE EXAMPLES FOR THE COMPUT A TION OF PERSISTENT HOMOLOGY 13 5. A worst-case example which is a flag complex In this section we show that after mo difying the split complex b y sub dividing some edges and triangles, the resulting filtered complex is the flag complex of a filtered graph. A t the end of the section we also show that these mo dified strip examples can b e realized as the Vietoris–Rips complex of a finite data cloud for some increasing sequence of real n um b ers. Definition 5.1. Let G = ( V , E ) b e an undirected lo op-free graph. The clique c omplex of G is a simplicial complex C l ( G ) whose v ertex set is V and a subset of v ertices { v 0 , . . . , v k } is a simplex in C l ( G ) if and only if for every i < j , { v i , v j } is an edge in G . Note that if G is a filtered graph with filtration ∅ ⊆ G 1 ⊆ G 2 ⊆ · · · ⊆ G m , the asso ciated sequence of clique complexes ∅ ⊆ C l ( G 1 ) ⊆ C l ( G 2 ) ⊆ · · · ⊆ C l ( G m ) gives a filtered simplicial complex. As defined, the split worst-case examples cannot be realized as clique complexes of filtered graphs since in the filtration the triangles app ear several steps after its b oundary edges. T o realize them as clique complexes, we alter the split examples as follows: 1 1 2 2 5 5 3 3 n n n − 1 n-1 n − 2 n-2 n + 1 = 0 ′ ( n − 2) ′ ( n − 2) ′′ 3 ′ 3 ′′ 2 ′ 2 ′′ 1 ′ 1 ′′ 5 ′ 5 ′′ n ′ n ′′ ( n − 1) ′ ( n − 1) ′′ • • • • • • − n ′′ − n ′ − n ∗ − n -n f n g n • . . . . . . • v n − 4 v n − 2 v n +1 v n − 1 v n − 3 v 4 v 2 v 1 v 3 v 5 v 7 v n v n +2 w n w 3 w 2 w 1 w 5 The picture is similar to the previous example; the difference is that w e add vertices w 1 , . . . , w n as the midpoints of the horizon tal base edges 1 ′ , . . . , n ′ , and denote the result- ing half edges as i ′ and i ′′ for i = 1 , . . . , n . Similarly , we add v ertices g 1 , . . . , g n at the midp oin ts of the edges − 1 , . . . , − n of the fin triangles and denote the resulting half edges as − i ′ and − i ′′ for i = 1 , . . . , n . W e also add new edges that divide the base and fin triangles into half. F or base triangles, these are denoted by e b 1 , . . . , e b n and for the fin triangles, by e f 1 , . . . , e f n . They are indexed by the triangles that they sub divide. F or the divided triangles, we denote the left one as b oxed i and the righ t one as the circled i , for base triangles. They are denoted b y b oxed − i and circled − i for fin triangles. W e ha ve the following filtration for this mo dified simplicial complex: (1) F in v ertices f 1 , . . . , f n , and g 1 , . . . , g n , (2) B ase v ertices v 1 , . . . , v n +2 , and w 1 , . . . , w n , (3) Horizontal base edges n ′′ , . . . , 1 ′′ , and n ′ , . . . , 1 ′ , (4) B ase edge n + 1 which is also lab eled as 0 ′ , (5) F in edges that destroy fin comp onents − n ∗ , . . . , − 1 ∗ , (6) F in edges that create cycles − n ′′ , . . . , − 1 ′′ , and − n ′ , . . . , − 1 ′ (7) V ertical base edges n, n − 1 , . . . , 1 , (8) B ase midlines and base triangles e b 1 , 1 , 1 , . . . , e b n , n , n , (9) Fin midlines and fin triangles e f 1 , − 1 , -1 , . . . , e f n , − n , -n . 14 UZA Y ÇETIN AND ERGÜN Y ALÇIN Note that for all i , the edges e b i and the base triangles i and i app ear at the same time in the filtration. Similarly for the fin triangles, the edges e f i and the fin triangles − i and i app ear at the same filtration lev el. W e call this mo dified example the mo difie d split example and denote it by Y ( n ) . F or each k = 0 , 1 , 2 , let B ′ ( k ) denote the b oundary matrix for ∂ k : C k ( Y ( n )) → C k − 1 ( Y ( n )) . F or the b oundary matrix, we use the linear ordering as giv en in the ab ov e list. Example 5.2. In this example w e illustrate how the b oundary matrix B ′ (2) for the mo dified split example Y (2) is reduced using the reduction algorithm. W e explain the pro cessing of the columns using the sparse matrix notation: 1 → ( e b 1 , 1 , 1 ′ ) d 1 = 1 + 1 → ( e b 1 , 1 , 1 ′ ) + ( e b 1 , 2 , 1 ′′ ) → (1 , 2 , 1 ′ , 1 ′′ ) So the triangle 1 is paired with e b 1 , and the triangle 1 is paired with edge 1 . F or the triangles 2 and 2 , w e hav e 2 → ( e b 2 , 0 ′ , 2 ′ ) 2 + 2 → ( e b 2 , 0 ′ , 2 ′ ) + ( e b 2 , 1 , 2 ′′ ) → (1 , 0 ′ , 2 ′ , 2 ′′ ) d 2 = ( 2 + 2 ) + d 1 → (1 , 0 ′ , 2 ′ , 2 ′′ ) + (1 , 2 , 1 ′ , 1 ′′ ) = (2 , 0 ′ , 1 ′ , 1 ′′ , 2 ′ , 2 ′′ ) . This shows that the triangle 2 is paired with e b 2 and the triangle 2 is paired with 2 . F or the fin triangles we ha ve a similar pairing pro cess. − 1 → ( e f 1 , − 1 ′ , − 1 ∗ ) − 1 + -1 → ( e f 1 , − 1 ′ , − 1 ∗ ) + ( e f 1 , 1 , − 1 ′′ ) → (1 , − 1 ′ , − 1 ′′ , − 1 ∗ ) ( − 1 + -1 ) + d 1 → (1 , − 1 ′ , − 1 ′′ , − 1 ∗ ) + (1 , 2 , 1 ′ , 1 ′′ ) → (2 , − 1 ′ , − 1 ′′ , − 1 ∗ , 1 ′ , 1 ′′ ) ( − 1 + -1 + d 1 ) + d 2 → (2 , − 1 ′ , − 1 ′′ , − 1 ∗ , 1 ′ , 1 ′′ ) + (2 , 0 ′ , 1 ′ , 1 ′′ , 2 ′ , 2 ′′ ) → ( − 1 ′ , − 1 ′′ , − 1 ∗ , 0 ′ , 2 ′ , 2 ′′ ) . This pairs the fin triangle − 1 with e f 1 and the fin triangle -1 with the edge − 1 ′ . Finally w e hav e − 2 → ( e f 2 , − 2 ′ , − 2 ∗ ) − 2 + -2 → ( e f 2 , − 2 ′ , − 2 ∗ ) + ( e f 2 , 2 , − 2 ′′ ) → (2 , − 2 ′ , − 2 ′′ , − 2 ∗ ) ( − 2 + -2 ) + d 2 → (2 , − 2 ′ , − 2 ′′ , − 2 ∗ ) + (2 , 0 ′ , 1 ′ , 1 ′′ , 2 ′ , 2 ′′ ) → ( − 2 ′ , − 2 ′′ , − 2 ∗ , 0 ′ , 1 ′ , 1 ′′ , 2 ′ , 2 ′′ ) . This pairs the triangle − 2 with e f 2 and the triangle -2 with the edge − 2 ′ . So the p ersisten t pairs for Y (2) for the 1-dimensional homology H 1 are { ( e b 1 , 1 ) , (1 , 1 ) , ( e b 2 , 2 ) , (2 , 2 ) , ( e f 1 , − 1 ) , ( − 1 ′ , -1 ) , ( e f 2 , − 2 ) , ( − 2 ′ , -2 ) } . Since some of the pairs consist of simplices that app ear at the same filtration level, they will app ear as p oints on the diagonal line in the p ersisten t diagram. No w we are ready to prov e the following: WORST-CASE EXAMPLES FOR THE COMPUT A TION OF PERSISTENT HOMOLOGY 15 Prop osition 5.3. The mo difie d strip example Y ( n ) is the clique c omplex of a filter e d gr aph and it has time c omplexity Ω( M 3 ) wher e M is the numb er of simplic es. Pr o of. A t each filtration lev el the mo dified strip example is the clique complex of its one sk eleton b ecause all the triangles in the complex appear at the same filtration level as the b oundary edges of the triangles. T o see that the reduction algorithm applied to B ′ ( n ) has cubic runtime complexit y , observe that the complex Y ( n ) has 4 n + 2 v ertices, 8 n + 1 edges, and 4 n triangles. So the num ber of simplices in Y ( n ) is M = 16 n + 3 . Note that in the reduction algorithm, the b oxed triangles are alwa ys paired with the edge that divides the triangle in to t w o. T o obtain th e c hains t i = i + i and t − i = − i + -i w e p erform 2 n v ector additions which is p erformed using 12 n field additions. The pro cessing of the chains t i and t − i are very similar to the pro cessing of the triangles in the original strip example. In fact w e use the same num b er of v ector additions to pro cess these chains. The only difference is that the v ectors added ha v e more nonzero en tries since more edges are introduced in the mo dified case. This increases the num ber of field additions in the reduction pro cess. W e now calculate exact num b er of fields additions used to pro cess columns for the b oundary matrix B ′ ( n ) . Let d i = t 1 + · · · + t i . Note that the columns for d i ha v e 2 i + 2 nonzero entries. Since the column for d i is obtained b y adding the columns t i and d i − 1 , the num ber of field additions to pro cess all the columns for t 1 , . . . , t n is (4 + 4) + (6 + 4) + (8 + 4) + · · · + (2 n + 4) = n 2 + 5 n − 6 . T o pro cess the column for t − j , we p erform the additions t − j + d j + · · · + d n whic h requires 4 + n X i = j (2 i + 2) = n 2 + 3 n − j 2 − j + 6 field additions. Adding these ov er all j = 1 , . . . , n , we see that to pro cess the columns for fin triangles, w e p erform 2 3 n ( n 2 + 3 n + 8) additions. Hence the total num b er of field additions to reduce the matrix B ′ (2) is S ′ ( n ) = 1 3 (2 n 3 + 9 n 2 + 37 n − 18) . Since M = 16 n + 3 , we ha ve S ′ ( n ) = Ω( M ) . Note that b oth the num ber of simplices and the n um b er of field additions in the reduction process increases roughly by a factor of 2 for the mo dified complex Y ( n ) compared to the reduction pro cess for the strip example X ( n ) . □ Next we sho w that the mo dified strip examples can b e realized as the Vietoris–Rips complex of a finite data cloud for a sequence of real n um b er 0 < r 1 < · · · < r s as scale parameters. W e first recall the definition of a Vietoris–Rips complex of a data cloud. Definition 5.4. Let X b e a finite set of p oints in R l . F or a real n umber r ≥ 0 , the Vietoris– R ips c omplex V R ( X , r ) is defined as the simplicial complex whose vertex set is X , and whose n -simplices are the subsets { x 0 , . . . , x n } of X such that d ( x i , x j ) ≤ r for every i, j . F or an increasing sequence 0 < r 1 < r 2 < · · · < r m of p ositive real num bers, w e can define a filtered simplicial complex by taking the simplicial complex K i to b e the Vietoris–Rips complex V R ( X , r i ) . The follo wing prop osition is implicit in classical distance geometry and the theory of Vietoris–Rips complexes, but w e w ere unable to find a direct reference. W e include a pro of here for completeness. 16 UZA Y ÇETIN AND ERGÜN Y ALÇIN Prop osition 5.5. L et ∅ ⊆ G 1 ⊆ · · · ⊆ G m b e a filter e d gr aph on a finite vertex set V = { 1 , . . . , n } , and let Cl( G t ) denote the clique c omplex of G t for t = 1 , . . . , m . Then ther e exist p oints X = { p 1 , . . . , p n } ⊂ R l and incr e asing se quenc e of r e al numb ers 0 < r 1 < · · · < r m such that for e ach t = 1 , . . . , m , ther e is an isomorphism of simplicial c omplexes V R ( X , r t ) ∼ = Cl( G t ) . Pr o of. It is well-kno wn that Vietoris–Rips complexes are flag complexes, hence they are clique complexes of underlying 1-dimensional sub complexes (see for example [7, Section I I I.2]). Therefore it suffices to realize the filtered graphs G i as nested distance–threshold graphs. Since the n = 1 case is trivial, we can assume n ≥ 2 . F or each unordered pair { i, j } , define its app e ar anc e level ℓ ( i, j ) = min { t : { i, j } ∈ E ( G t ) } , with ℓ ( i, j ) = ∞ if the edge never app ears. Note that ℓ ( i, j ) ≥ 1 for all i, j . Fix ε := 1 n − 1 , δ t := 2 − t ε for t = 1 , . . . , m , and let δ m +1 := 0 . Then w e hav e 1 2 ≥ δ 1 > · · · > δ m > δ m +1 = 0 and for eac h i , X j  = i δ ℓ ( i,j ) ≤ ( n − 1) δ 1 = ( n − 1) ϵ 2 = 1 2 . Define an n × n symmetric matrix K = ( K ij ) b y K ii = 1 , K ij = δ ℓ ( i,j ) for all i  = j. By Gershgorin’s circle theorem (see, for example, [14]), every eigenv alue of K lies in the in terv al [1 − 1 2 , 1 + 1 2 ] . In particular K is p ositiv e definite. By a standard result in linear algebra, every real symmetric p ositive definite matrix is a Gram matrix. Thus there exist vectors p 1 , . . . , p n ∈ R l suc h that K ij = ⟨ p i , p j ⟩ (see [1, Theorem 7.35]). In particular, ∥ p i ∥ = 1 for all i , and ∥ p i − p j ∥ 2 = ∥ p i ∥ 2 + ∥ p j ∥ 2 − 2 ⟨ p i , p j ⟩ = 2 − 2 δ ℓ ( i,j ) . F or each t = 1 , . . . , m + 1 , define d t := p 2 − 2 δ t . F or all i, j , we ha ve ∥ p i − p j ∥ = d ℓ ( i,j ) . Since the sequence δ t is strictly decreasing, the sequence d t is strictly increasing. W e ha v e 1 ≤ d 1 < d 2 < · · · < d m < d m +1 = √ 2 . F or each t = 1 , . . . , m , c ho ose a radius r t suc h that d t < r t < d t +1 . Then ∥ p i − p j ∥ ≤ r t ⇐ ⇒ d ℓ ( i,j ) ≤ d t ⇐ ⇒ ℓ ( i, j ) ≤ t ⇐ ⇒ { i, j } ∈ E ( G t ) . Hence the distance–threshold graph at scale r t is exactly G t . This completes the pro of. □ W e finish the pap er with the following corollary . WORST-CASE EXAMPLES FOR THE COMPUT A TION OF PERSISTENT HOMOLOGY 17 Corollary 5.6. The mo difie d strip example Y ( n ) c an b e r e alize d as the Vietoris–Rips c omplex of a finite data set for some se quenc e of r e al numb ers 0 < r 1 < · · · < r m as sc ale p ar ameters. Pr o of. This follows from Prop ositions 5.3 and 5.5. □ W orst-Case examples for Erdös-Rényi and Vietoris-Rips filtration mo dels w ere also given in [9]. F or these mo dels the w orst-case runtime is Θ( n 7 ) since the pro cessing of the all higher dimensional simplices is also considered. A ckno wledgements and Funding Declara tion The second author is supp orted by TÜBİT AK 2219-In ternational Postdoctoral Researc h F ellowship Program (2023, 2nd term). W e gratefully ac kno wledge TÜBİT AK for its supp ort of this researc h. W e thank Matthew Gelvin, Barış Coşkunüzer, Caroline Y alçın, and Kora y Karabina for reading an earlier v ersion of the pap er and for their helpful comments. Part of the pap er was written while the second author w as on sabbatical leav e visiting Universit y of W aterlo o. The second author thanks Univ ersity of W aterlo o for hospitalit y during his sabbatical lea ve. References [1] Sheldon Axler. Line ar Algebr a Done Right . Springer, 3 edition, 2015. [2] Gunnar Carlsson. T op ology and data. Bul letin of the Americ an Mathematic al So ciety , 46:255–308, 2009. [3] F rédéric Chazal and Bertrand Michel. An introduction to top ological data analysis: F undamental and practical asp ects for data scien tists. F r ontiers in Artificial Intel ligenc e , 4:667963, 2021. [4] Chao Chen and Michael Kerb er. Persisten t homology computation with a twist. In Pr o c e e dings of the 27th Eur op e an W orkshop on Computational Geometry (Eur oCG 2011) , pages 1–4, 2011. [5] Baris Coskunuzer and Cüneyt Gürcan Akçora. T op ological metho ds in machine learning: A tutorial for practitioners, 2024. [6] T amal K. Dey and Y usu W ang. Computational T opolo gy for Data Analysis . Cam bridge Universit y Press, 2022. [7] Herb ert Edelsbrunner and John Harer. Computational T op olo gy: An Intr o duction . American Mathemat- ical So ciet y , 2010. [8] Herb ert Edelsbrunner, David Letscher, and Afra Zomoro dian. T op ological p ersistence and simplification. Discr ete & Computational Ge ometry , 28:511–533, 2002. [9] Barbara Giun ti, Guillaume Houry , and Michael Kerber. A verage complexity of matrix reduction for clique filtrations. In Pr o c e e dings of the 2022 International Symp osium on Symb olic and Algebr aic Computation , ISSA C ’22, page 187–196, New Y ork, NY, USA, 2022. Asso ciation for Computing Machinery . [10] Dmitriy Morozov. Persistence algorithm takes cubic time in the w orst case. BioGeometry News, F ebruary 2005. Departmen t of Computer Science, Duke Universit y . [11] Dmitriy Morozo v and Primoz Skraba. Persisten t (co)homology in matrix multiplication time, 2024. [12] Nina Otter, Mason A. Porter, Ulrike Tillmann, P eter Grindro d, and Heather A. Harrington. A roadmap for the computation of persistent homology . EPJ Data Scienc e , 6(1):17, 2017. [13] Raul Rabadán and Andrew J. Blumberg. T op olo gic al Data A nalysis for Genomics and Evolution: T op olo gy in Biolo gy . Cambridge Universit y Press, Cambridge and New Y ork, 2020. [14] Thomas S. Shores. Applie d Line ar Algebr a and Matrix Analysis . Undergraduate T exts in Mathematics. Springer, 2 edition, 2018. [15] Afra Zomoro dian and Gunnar Carlsson. Computing p ersistent homology . Discr ete & Computational Ge- ometry , 33:249–274, 2005. Appendix A. An algorithm for constructing the strip examples An algorithm to construct the filtered simplicial complex strip X ( n ) for n ≥ 2 can b e giv en as follows. The complex X (1) can b e constructed easily b y inserting the list of simplices from the picture. 18 UZA Y ÇETIN AND ERGÜN Y ALÇIN Algorithm 4 Building ordered lists of v ertices, edges, and triangles of the simplicial complex X ( n ) for n ≥ 2 1: function BuildVer tices ( n ) 2: return [ f 1 , . . . , f n , v 1 , . . . , v n +2 ] 3: end function 4: function BuildEdges ( n ) 5: E ← [ ] ▷ Horizon tal base edges n ′ , . . . , 1 ′ , 0 ′ 6: for i = n, n − 1 , . . . , 3 do 7: app end ( v i − 2 , v i +2 ) to E ▷ edge i ′ 8: end for 9: app end ( v 1 , v 4 ) to E ▷ edge 2 ′ 10: app end ( v 2 , v 3 ) to E ▷ edge 1 ′ 11: app end ( v n , v n +2 ) to E ▷ edge 0 ′ ▷ Fin edges − n ∗ , . . . , − 1 ∗ 12: for i = n, n − 1 , . . . , 2 do 13: app end ( f i , v i − 1 ) to E ▷ edge − i ∗ 14: end for 15: app end ( f 1 , v 1 ) to E ▷ edge − 1 ∗ ▷ Fin edges − n, . . . , − 1 16: for i = n, n − 1 , . . . , 1 do 17: app end ( f i , v i +1 ) to E ▷ edge − i 18: end for ▷ V ertical base edges n, . . . , 1 19: for i = n, n − 1 , . . . , 2 do 20: app end ( v i − 1 , v i +1 ) to E ▷ edge i 21: end for 22: app end ( v 1 , v 2 ) to E ▷ edge 1 23: return E 24: end function 25: function BuildTriangles ( n ) 26: T ← [ ] ▷ Base triangles 1 , 2 , . . . , n 27: app end ( v 1 , v 2 , v 3 ) to T ▷ triangle 1 28: app end ( v 1 , v 2 , v 4 ) to T ▷ triangle 2 29: for i = 3 , . . . , n do 30: app end ( v i − 2 , v i , v i +2 ) to T ▷ triangle i 31: end for ▷ Fin triangles -1 , -2 , . . . , -n 32: app end ( f 1 , v 1 , v 2 ) to T ▷ triangle -1 33: for i = 2 , . . . , n do 34: app end ( f i , v i − 1 , v i +1 ) to T ▷ triangle -i 35: end for 36: return T 37: end function WORST-CASE EXAMPLES FOR THE COMPUT A TION OF PERSISTENT HOMOLOGY 19 Algorithm 5 Constructing the filtered simplicial complex X ( n ) for n ≥ 2 1: function Crea teWorstCaseComplex ( n ) 2: V ← BuildVer tices ( n ) 3: E ← BuildEdges ( n ) 4: T ← BuildTriangles ( n ) 5: K ← [ ] 6: t ← 1 7: for each v ∈ V do 8: app end ( { v } , t ) to K 9: t ← t + 1 10: end for 11: for each e ∈ E do 12: app end (sort( e ) , t ) to K 13: t ← t + 1 14: end for 15: for each σ ∈ T do 16: app end (sort( σ ) , t ) to K 17: t ← t + 1 18: end for 19: return K 20: end function Uza y Çetin, Dep ar tment of Ma thema tics, The Ohio St a te University, Columbus, OH, 43210- 1174, USA Email addr ess : cetin.16@osu.edu Er gün Y alçın, Dep ar tment of Ma thema tics, Bilkent University, 06800 Bilkent, Ankara, Turkiye Email addr ess : yalcine@fen.bilkent.edu.tr