A Framework for In-place Graph Algorithms

Read-only memory (ROM) model is a classical model of computation to study time-space tradeoffs of algorithms. A classical result on the ROM model is that any algorithm to sort n numbers using O(s) words of extra space requires Omega (n^2/s) comparisons for lg n <= s <= n/lg n and the bound has also been recently matched by an algorithm. However, if we relax the model, we do have sorting algorithms (say Heapsort) that can sort using O(n lg n) comparisons using O(lg n) bits of extra space, even keeping a permutation of the given input sequence at anytime during the algorithm. We address similar relaxations for graph algorithms. We show that a simple natural relaxation of ROM model allows us to implement fundamental graph search methods like BFS and DFS more space efficiently than in ROM. By simply allowing elements in the adjacency list of a vertex to be permuted, we show that, on an undirected or directed connected graph G having n vertices and m edges, the vertices of G can be output in a DFS or BFS order using O(lg n) bits of extra space and O(n^3 lg n) time. Thus we obtain similar bounds for reachability and shortest path distance (both for undirected and directed graphs). With a little more (but still polynomial) time, we can also output vertices in the lex-DFS order. As reachability in directed graphs (even in DAGs) and shortest path distance (even in undirected graphs) are NL-complete, and lex-DFS is P-complete, our results show that our model is more powerful than ROM if L != P. En route, we also introduce and develop algorithms for another relaxation of ROM where the adjacency lists of the vertices are circular lists and we can modify only the heads of the lists. Here we first show a linear time DFS implementation using n + O(lg n) bits of extra space. Improving the extra space exponentially to only O(lg n) bits, we also obtain BFS and DFS albeit with a slightly slower running time. Both the models we propose maintain the graph structure throughout the algorithm, only the order of vertices in the adjacency list changes. In sharp contrast, for BFS and DFS, to the best of our knowledge, there are no algorithms in ROM that use even O(n^{1-epsilon}) bits of extra space; in fact, implementing DFS using cn bits for c<1 has been mentioned as an open problem. Furthermore, DFS (BFS, respectively) algorithms using n+o(n) (o(n), respectively) bits of extra use Reingold\u27s [JACM, 2008] or Barnes et al\u27s reachability algorithm [SICOMP, 1998] and hence have high runtime. Our results can be contrasted with the recent result of Buhrman et al. [STOC, 2014] which gives an algorithm for directed st-reachability on catalytic Turing machines using O(lg n) bits with catalytic space O(n^2 lg n) and time O(n^9)

Space-Efficient Algorithms for Longest Increasing Subsequence

Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in O(n log n) time and space. Our goal in this paper is to reduce the space consumption while keeping the time complexity small. For sqrt(n) <= s <= n, we present algorithms that use O(s log n) bits and O(1/s n^2 log n) time for computing the length of a longest increasing subsequence, and O(1/s n^2 log^2 n) time for finding an actual subsequence. We also show that the time complexity of our algorithms is optimal up to polylogarithmic factors in the framework of sequential access algorithms with the prescribed amount of space

The Heaviest Induced Ancestors Problem Revisited

We revisit the heaviest induced ancestors problem, which has several interesting applications in string matching. Let T_1 and T_2 be two weighted trees, where the weight W(u) of a node u in either of the two trees is more than the weight of u\u27s parent. Additionally, the leaves in both trees are labeled and the labeling of the leaves in T_2 is a permutation of those in T_1. A node x in T_1 and a node y in T_2 are induced, iff their subtree have at least one common leaf label. A heaviest induced ancestor query HIA(u_1,u_2) is: given a node u_1 in T_1 and a node u_2 in T_2, output the pair (u_1^*,u_2^*) of induced nodes with the highest combined weight W(u^*_1) + W(u^*_2), such that u_1^* is an ancestor of u_1 and u^*_2 is an ancestor of u_2. Let n be the number of nodes in both trees combined and epsilon >0 be an arbitrarily small constant. Gagie et al. [CCCG\u27 13] introduced this problem and proposed three solutions with the following space-time trade-offs: - an O(n log^2n)-word data structure with O(log n log log n) query time - an O(n log n)-word data structure with O(log^2 n) query time - an O(n)-word data structure with O(log^{3+epsilon}n) query time. In this paper, we revisit this problem and present new data structures, with improved bounds. Our results are as follows. - an O(n log n)-word data structure with O(log n log log n) query time - an O(n)-word data structure with O(log^2 n/log log n) query time. As a corollary, we also improve the LZ compressed index of Gagie et al. [CCCG\u27 13] for answering longest common substring (LCS) queries. Additionally, we show that the LCS after one edit problem of size n [Amir et al., SPIRE\u27 17] can also be reduced to the heaviest induced ancestors problem over two trees of n nodes in total. This yields a straightforward improvement over its current solution of O(n log^3 n) space and O(log^3 n) query time

String Periods in the Order-Preserving Model

The order-preserving model (op-model, in short) was introduced quite recently but has already attracted significant attention because of its applications in data analysis. We introduce several types of periods in this setting (op-periods). Then we give algorithms to compute these periods in time O(n), O(n log log n), O(n log^2 log n/log log log n), O(n log n) depending on the type of periodicity. In the most general variant the number of different periods can be as big as Omega(n^2), and a compact representation is needed. Our algorithms require novel combinatorial insight into the properties of such periods

Stability Yields Sublinear Time Algorithms for Geometric Optimization in Machine Learning

In this paper, we study several important geometric optimization problems arising in machine learning. First, we revisit the Minimum Enclosing Ball (MEB) problem in Euclidean space ?^d. The problem has been extensively studied before, but real-world machine learning tasks often need to handle large-scale datasets so that we cannot even afford linear time algorithms. Motivated by the recent developments on beyond worst-case analysis, we introduce the notion of stability for MEB, which is natural and easy to understand. Roughly speaking, an instance of MEB is stable, if the radius of the resulting ball cannot be significantly reduced by removing a small fraction of the input points. Under the stability assumption, we present two sampling algorithms for computing radius-approximate MEB with sample complexities independent of the number of input points n. In particular, the second algorithm has the sample complexity even independent of the dimensionality d. We also consider the general case without the stability assumption. We present a hybrid algorithm that can output either a radius-approximate MEB or a covering-approximate MEB, which improves the running time and the number of passes for the previous sublinear MEB algorithms. Further, we extend our proposed notion of stability and design sublinear time algorithms for other geometric optimization problems including MEB with outliers, polytope distance, one-class and two-class linear SVMs (without or with outliers). Our proposed algorithms also work fine for kernels