All-Pairs Min-Cut in Sparse Networks

Algorithms are presented for the all-pairs min-cut problem in bounded treewidth, planar, and sparse networks. The approach used is to preprocess the input n-vertex network so that afterward, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute min-cuts subsequently. In particular, after an Onlog Ž n. preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for Ž 2 such networks the all-pairs min-cut problem can be solved in time On.. This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, �, of the input network. The parameter � varies between 1 and �Ž. n; the algorithms perform well when � � on. Ž. The value Ž 2 of a min-cut can be found in time On� � log �. and all-pairs min-cut can be Ž 2 4 solved in time On � � log �. for sparse networks. The corresponding runnin

Prefix Graphs and Their Applications

The range product problem is, for a given set S equipped with an associative operator ffi, to preprocess a sequence a1 ; : : : ; an of elements from S so as to enable efficient subsequent processing of queries of the form: Given a pair (s; t) of integers with 1 s t n, return as ffi as+1 ffi \Delta \Delta \Delta ffi a t . The generic range product problem and special cases thereof, usually with ffi computing the maximum of its arguments according to some linear order on S, have been extensively studied. We show that a large number of previous sequential and parallel algorithms for these problems can be unified and simplified by means of prefix graphs

Sensitive Functions and Approximate Problems

We investigate properties of functions that are good measures of the CRCW PRAM complexity of comput-ing them. While the block sensitivity is known to be a good measure of the CREW PRAM complexity, no such measure is known for CRCW PRAMs. We show that the complexity of computing a function is related to its everywhere sensitivity, introduced by Vishkin and Wigderson. Specifically we show that the time required to compute a function f: D "

Tight Bounds on Oblivious Chaining

The chaining problem is defined as follows. Given values a 1 ; :::; an ; a i = 0 or 1, 1 i n, compute b 1 ; :::; b n , such that b i = maxfj j a j = 1; j ! ig. (Define maxfg = 0) The chaining problem appears as a subproblem in many contexts. There are algorithms known that solve the chaining problem on CRCW PRAMs in O(ff(n)) time, where ff(n) is the inverse of Ackerman's function, and is a very slowly growing function. We study a class of algorithms (called oblivious algorithms) for this problem. We present a simple oblivious chaining algorithm running in O(ff(n)) time. More importantly, we demonstrate the optimality of the algorithm by showing a matching lower bound for oblivious algorithms using n processors. We also provide the first steps towards a lower bound for all chaining algorithms by showing that any chaining algorithm that runs in two steps must use a superlinear number of processors. Our proofs use prefix graphs and weak superconcentrators. We demonstrate an interestin..

Deterministic restrictions in circuit complexity

We study the complexity of computing Boolean functions using AND, OR and NOT gates. We show that a circuit of depth d with S gates can be made to output a constant by setting O(S 1−ɛ(d) ) (where ɛ(d) = 4 −d) of its input values. This implies a superlinear size lower bound for a large class of functions. Using this, we obtain a function computable by a uniform family of constant depth polynomial size circuits that cannot be computed by constant depth circuits of linear size. We give circuit constructions that show that the bound O(S 1−ɛ(d) ) is near optimal. We also study the complexity of computing threshold functions. The function T n r has the value 1 iff at least r of its inputs have the value 1. We show that a circuit computing T n r has at least Ω(r 2 (log n) / log r) gates, for r ≤ n 1/3, improving previous bounds. We also show a trade-off between the number of gates and the number of wires in a threshold circuit, namely, a circuit with G (< n/2) gates and W wires computing T n r satisfies W ≥ Ω(nr(log n)/(log(G / log n))), showing that it is not possible to simultaneously optimize the number of gates and wires in a threshold circuit. Our bounds for threshold functions are based on a combinatorial lemma of independent interest.

Shortest Paths in Digraphs of Small Treewidth. Part II: Optimal Parallel Algorithms

We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give parallel algorithms for the EREWPRAM model of computation that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(ff(n)) time using a single processor, after a preprocessing of O(log 2 n) time and O(n) work, where ff(n) is the inverse of Ackermann's function. The class of constant treewidth graphs contains outerplanar graphs and seriesparallel graphs, among others. To the best of our knowledge, these are the first parallel algorithms which achieve these bounds for any class of graphs except trees. We also give a dynamic algorithm which, after a change in an edge weight, updates our data structures in O(log n) time using O(n fi ) work, for any constant 0 ! fi ! 1. Moreover, we give an algorithm of independ..

The Complexity of Parallel Prefix Problems on Small Domains

We show non-trivial lower bounds for several prefix problems in the CRCW PRAM model. The chaining problem is, given a binary input, for each 1 in the input, find the index of the nearest 1 (1) to its left. Our main result is that for an input of n bits, solving the chaining problem using O(n) processors requires inverse-Ackerman time. We give a reduction to show that the same lower bound applies to a parenthesis matching problem, again matching the previously known upper bound. We also give reductions to show that similar lower bounds hold for the prefix maxima and the range maxima problems. 1 Introduction Lower bounds in parallel computation often depend critically on the domain size of the problem that is being solved. Typically, these lower bounds use Ramsey theoretic arguments to force the algorithms to behave in a structured manner on some subset of the inputs. Then, it is argued that this subset of inputs (3,4) is rich enough that this structured behaviour cannot find a quic..

Shortest Paths in Digraphs of Small Treewidth. Part II: Optimal Parallel Algorithms

We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give parallel algorithms for the EREWPRAM model of computation that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(ff(n)) time using a single processor, after a preprocessing of O(log² n) time and O(n) work, where ff(n) is the inverse of Ackermann's function. The class of constant treewidth graphs contains outerplanar graphs and seriesparallel graphs, among others. To the best of our knowledge, these are the first parallel algorithms which achieve these bounds for any class of graphs except trees. We also give a dynamic algorithm which, after a change in an edge weight, updates our data structures in O(log n) time using O(n fi ) work, for any constant 0 ! fi ! 1. Moreover, we give an algorithm ..

Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms

We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(ff(n)) time after O(n) preprocessing. This improves upon previously known results for the same problem. We also give a dynamic algorithm which, after a change in an edge weight, updates the data structure in time O(n fi ), for any constant 0 ! fi ! 1. Furthermore, an algorithm of independent interest is given: computing a shortest path tree, or finding a negative cycle in linear time