범위 분할 엔트로피 기반 알고리즘의 보편 최적성 증명
📝 Abstract
TimSort is a well-established sorting algorithm whose running time depends on how sorted the input already is. Recently, Eppstein, Goodrich, Illickan, and To designed algorithms inspired by TimSort for Pareto front, planar convex hull, and two other problems. For each of these problems, they define a Range Partition Entropy; a function H mapping lists I that store n points to a number between 0 and log n. Their algorithms have, for each list of points I, a running time of O(n(1 + H(I))). In this paper, we provide matching lower bounds for the Pareto front and convex hull algorithms by Eppstein, Goodrich, Illickan, and To. In particular, we show that their algorithm does not correspond to TimSort (or related stack-based MergeSort variants) but rather to a variant of QuickSort. From this, we derive an intuitive notion of universal optimality. We show comparisonbased lower bounds that prove that the algorithms by Eppstein, Goodrich, Illickan and To are universally optimal under this notion of universal optimality. ACM Subject Classification Theory of computation → Computational geometry; Theory of computation → Design and analysis of algorithms Keywords and phrases Convex hull, Combinatorial proofs, Universal optimality
💡 Analysis
TimSort is a well-established sorting algorithm whose running time depends on how sorted the input already is. Recently, Eppstein, Goodrich, Illickan, and To designed algorithms inspired by TimSort for Pareto front, planar convex hull, and two other problems. For each of these problems, they define a Range Partition Entropy; a function H mapping lists I that store n points to a number between 0 and log n. Their algorithms have, for each list of points I, a running time of O(n(1 + H(I))). In this paper, we provide matching lower bounds for the Pareto front and convex hull algorithms by Eppstein, Goodrich, Illickan, and To. In particular, we show that their algorithm does not correspond to TimSort (or related stack-based MergeSort variants) but rather to a variant of QuickSort. From this, we derive an intuitive notion of universal optimality. We show comparisonbased lower bounds that prove that the algorithms by Eppstein, Goodrich, Illickan and To are universally optimal under this notion of universal optimality. ACM Subject Classification Theory of computation → Computational geometry; Theory of computation → Design and analysis of algorithms Keywords and phrases Convex hull, Combinatorial proofs, Universal optimality
📄 Content
In better-than-worst-case analysis, algorithms are designed to be optimal with respect to structural properties of the input, not just its size. Classical examples include: output-sensitive algorithms [13], whose running time depends on n and the output size k; order-oblivious algorithms [1], which fix the input values and consider for an algorithm the worst-case running time over all input lists that contain these values; sorting algorithms that adapt to how presorted the input is [2,3,4,7,12,16,17,19].
Entropy. A powerful tool for proving better-than-worst-case running times is the use of entropy functions. Formally, an entropy function takes as input a list I of n values and outputs a number between 0 and log n. A classical example arises in the analysis of TimSort.
For the sorting problem, the input is a list I of values. We define an ordered partition Π = (I 1 , I 2 , . . .) as a partition of I into runs, where each run I i is a contiguous sublist of I whose values are either non-decreasing or non-increasing. This induces the TimSort entropy
Inspired by these two entropy measures, Eppstein, Goodrich, Illickan, and To [6] define entropy functions for a range of geometric problems, including Pareto front, planar convex hull, 3D convex hull, the lower envelope of simple polygonal chains, and the visibility polygon of a point inside a convex polygon with convex holes. Although their entropy definitions differ from problem to problem, they share a common structure. We illustrate their entropy function for constructing the planar convex hull of a size-n point set P given as a list I.
Let I be a list storing P . A respectful partition is a list Π = ((I 1 , R 1 ), . . .) of pairs (I i , R i ), where I i is a subsequence of I and R i is a triangle containing I i such that (i) the sets I 1 , I 2 , . . . form a partition of I, (ii) either I i is sorted (e.g., by x-coordinate) or R i is contained in the convex hull of P , and (iii) if R i is not contained in the convex hull of P (so I i is sorted), then R i is disjoint from all other triangles R j . This induces the range partition entropy [6] H(I) := min respectful partitions Π of I (Ii,Ri)∈Π
Eppstein, Goodrich, Illickan, and To adapt the algorithm of [13] to compute the convex hull of P in O(n(1 + H(I))) time. Every structural partition is also a respectful partition, and hence for all inputs I we have H(I) ≤ H s (I). These two quantities can be separated: for example, if all points of P lie on the convex hull but I lists all points of P by increasing x-coordinate, then H(I) = 0, and their algorithm uses linear time whilst [1] uses O(n log n).
Universal optimality. Entropy functions are appealing because they allow us to upper-bound the running time of an algorithm in a more fine-grained manner. Their true value, however, emerges when we can also provide matching lower bounds. A variety of frameworks have been developed for proving such better-than-worst-case lower bounds. Recently Haeupler, Wajc, and Zuzic [10, STOC ‘21] introduced universal optimality as an umbrella notion for these analyses, which has gained substantial traction [8,9,10,11,21,22,23,24,25].
Consider an algorithmic problem where U n denotes the universe of all possible inputs of size n. The usual worst-case running time of an algorithm A is max I∈Un Runtime(A, I). The core idea is to consider smaller universes U 1 , U 2 , . . . whose union covers U n . For a fixed universe U j and algorithm A, its universal running time is the worst-case running time over all inputs in U j . The universal lower bound is the best possible universal running time:
Universal(A, U j ) := max I∈Uj Runtime(A, I) and Universal-LB(U j ) := min A Universal(U j , A) An algorithm A is universally optimal if for all U j , Universal(A, U j ) ∈ O(Universal-LB(U j )).
As a concrete example, the notion of optimality for planar convex hull construction introduced by Afshani, Barbay, and Chan [1] fits naturally into this framework. For each size-n planar point set P , define the universe U P to consist of all lists I that store P . Clearly, the union of all such universes U P covers the universe of all size-n inputs for planar convex hull. For a fixed point set P , the universal running time is the worst-case running time of an algorithm over all possible orders in which the points of P may be presented. Afshani, Barbay, and Chan prove that the algorithm of [13] is universally optimal under this interpretation.
For sorting, one can define a universe U by fixing a sequence of sizes (s 1 , s 2 , . . .) that together sum up to n and assigning to it all inputs I for which there exists an ordered partition Π of I where each part I i ∈ Π has size s i . The better-than-worst-case optimality of TimSort corresponds to universal optimality under this interpretation.
Contribution. This paper revisits the analyses of Eppstein, Goodrich, Illickan, and To [6] for Pareto front and planar convex hulls. Consider the family I n of size-
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