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Fast ckr partitions of sparse graphs
 Chicago Journal of Theoretical Computer Science
"... We present fast algorithms for constructing probabilistic embeddings and approximate distance oracles in sparse graphs. The main ingredient is a fast algorithm for sampling the probabilistic partitions of Calinescu, Karloff, and Rabani in sparse graphs. 1 ..."
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We present fast algorithms for constructing probabilistic embeddings and approximate distance oracles in sparse graphs. The main ingredient is a fast algorithm for sampling the probabilistic partitions of Calinescu, Karloff, and Rabani in sparse graphs. 1
A Faster Algorithm for the Single Source Shortest Path Problem with Few Distinct Positive Lengths
"... In this paper, we propose an efficient method for implementing Dijkstra’s algorithm for the Single Source Shortest Path Problem (SSSPP) in a graph with positively lengthed edges, and where there are few distinct lengths. The SSSPP is one of the most widely studied problems in theoretical computer sc ..."
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In this paper, we propose an efficient method for implementing Dijkstra’s algorithm for the Single Source Shortest Path Problem (SSSPP) in a graph with positively lengthed edges, and where there are few distinct lengths. The SSSPP is one of the most widely studied problems in theoretical computer science and operations research. On a graph with n vertices, m edges and K distinct edge lengths, our algorithm runs in O(m) time if nK ≤ 2m and O(m log nK m) time, otherwise. We tested our algorithm against some of the fastest algorithms for SSSPP on arbitrarily (but positively) lengthed graphs. Our experiments on graphs with few edge lengths confirmed our theoretical results as the proposed algorithm consistently dominated the other SSSPP algorithms that did not exploit the special structure of having few distinct edge lengths.
Designing Communication Networks with Fixed or Nonblocking Traffic Requirements
, 1992
"... A general framework for specifying communication network design problems is given. We analyze the computational complexity of several specific problems within this framework. For fixed multirate traffic requirements, we prove that a particular network analysis problem is npcomplete, although severa ..."
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A general framework for specifying communication network design problems is given. We analyze the computational complexity of several specific problems within this framework. For fixed multirate traffic requirements, we prove that a particular network analysis problem is npcomplete, although several related network design problems are either efficiently solvable or have good approximation algorithms. For the case when we wish the network to operate without blocking any connection requests, we give efficient algorithms for dimensioning the link capacities of the network. This work is supported by the National Science Foundation, Bell Communications Research, Bell Northern Research, Digital Equipment Corporation, Italtel SIT, NEC, NTT, and SynOptics. 1. Introduction Much work has been done on the computational problem of designing lowcost communication networks (see [GN89, GTD + 89, GW90, GK90, KKG91, AKR91] and references therein). The general problem is: given a collection of no...
Shortest Paths in TimeDependent FIFO Networks Using Edge Load Forecasts
, 2009
"... We study the problem of finding shortest paths in timedependent networks with edge load forecasts where the behavior of each edge is modeled as a timedependent arrival function with FIFO property. Here, we present a new algorithm that computes for a given start node s and destination node d, the sh ..."
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We study the problem of finding shortest paths in timedependent networks with edge load forecasts where the behavior of each edge is modeled as a timedependent arrival function with FIFO property. Here, we present a new algorithm that computes for a given start node s and destination node d, the shortest paths and earliest arrival times for all possible starting times. Our algorithm runs in time O((Fd + λ)(E  + V log V )) where Fd is the output size (number of linear pieces needed to represent the earliest arrival time function) and λ is the input size (number of linear pieces needed to represent the local earliest arrival time functions for all edges in the network). Our method improves significantly on the best previously known algorithm which requires time O(FmaxV E) where Fmax ≥ Fd is the maximum number of linear pieces needed to represent the earliest arrival time function between the start node s to any node in the network. It has been conjectured that there are cases where Fmax is of superpolynomial size; however, even in such cases, Fd might still be of linear size. In such cases, our algorithm would take polynomial time to find the solution, while other methods require superpolynomial time. Both of the above methods are not useful in practice for graphs where Fd is of superpolynomial size. For such graphs, we present the first approximation method to compute for all possible starting times at s, the earliest arrival times at d within error at most ǫ. Our algorithm runs in time O ( ∆ (E  + V log V )) where ∆ is the difference beǫ tween the earliest arrival times at d for the latest and earliest starting times at s.
One Tree Suffices: A Simultaneous O(1)Approximation for SingleSink BuyatBulk
, 2012
"... We study the singlesink buyatbulk problem with an unknown cost function. We wish to route flow from a set of demand nodes to a root node, where the cost of routing x total flow along an edge is proportional to f (x) for some concave, nondecreasing function f satisfying f (0) = 0. We present a ..."
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We study the singlesink buyatbulk problem with an unknown cost function. We wish to route flow from a set of demand nodes to a root node, where the cost of routing x total flow along an edge is proportional to f (x) for some concave, nondecreasing function f satisfying f (0) = 0. We present a simple, fast, combinatorial algorithm that takes a set of demands and constructs a single tree T such that for all f the cost f (T) is a 47.45approximation of the optimal cost for that particular f. This is within a factor of 2.33 of the best approximation ratio currently achievable when the tree can be optimized for a specific function. Trees achieving simultaneous O(1)approximations for all concave functions were previously not known to exist regardless of computation time.
Parallel Algorithms for the k Shortest Paths and Related Problems
, 1996
"... A parallel algorithm is developed to find the k shortest paths between pairs of vertices in an edgeweighted directed graph. The concurrentread exclusivewrite PRAM is used as the model of computation. The algorithm computes an implicit representation of the k shortest paths to a given destination ..."
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A parallel algorithm is developed to find the k shortest paths between pairs of vertices in an edgeweighted directed graph. The concurrentread exclusivewrite PRAM is used as the model of computation. The algorithm computes an implicit representation of the k shortest paths to a given destination vertex from each vertex of a graph with n vertices and m edges, using O(m + nk log 2 k) work and O(log 3 k log k + log n(log log k + log n)) time, assuming that a shortest path tree rooted at the destination is precomputed. Parallel algorithms are also described for a weighted version of the problem of selecting an element of given rank from an unsorted array and for the selection of the kth smallest element in a matrix with sorted columns. The k shortest paths algorithm is applied to obtain a parallel implementation of a dynamic programming algorithm for the list Viterbi decoding problem, where one must find the most probable state sequences of a Markov process, given noisy obser...
Exercise 23.16 in [6].
, 2000
"... 2: Array initialization Section III.8.1 of [15] contains a description of how a bitvector can be intitialized in worst case constant time. ..."
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2: Array initialization Section III.8.1 of [15] contains a description of how a bitvector can be intitialized in worst case constant time.
SARA: Combining Stack Allocation and Register Allocation
"... Abstract. Commonlyused memory units enable a processor to load and store multiple registers in one instruction. We showed in 2003 how to extend gcc with a stacklocationallocation (SLA) phase that reduces memory traffic by rearranging the stack and replacing some load/store instructions with load/ ..."
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Abstract. Commonlyused memory units enable a processor to load and store multiple registers in one instruction. We showed in 2003 how to extend gcc with a stacklocationallocation (SLA) phase that reduces memory traffic by rearranging the stack and replacing some load/store instructions with load/storemultiple instructions. While speeding up the target code, our technique leaves room for improvement because of the phase ordering of register allocation before SLA. In this paper we present SARA which combines SLA and register allocation into a single phase. SARA creates a synergy among register assignment, spillcode generation, and SLA that makes the combined phase generate faster code than a sequence of the individual phases. We specify SARA by an integer linear program generated from the program text. We have implemented SARA in gcc, replacing gcc’s own implementation of register allocation. For our benchmarks, our results show that the target code is up to 16% faster than gcc with a separate SLA phase. 1
Literature Notes on Homeworks and the Takehome Exam
, 2000
"... lgorithm Exercise 2 of homework 4 in [13]. Prim's algorithm for computing a minimum spanning tree is, e.g., described in Section 24.2 of [6]. 5: The Stable Marriage Problem Exercise 1 of homework 6 in [13]. Gale and Shapley were the rst to investigate the stable marriage problem and gave an O(n ..."
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lgorithm Exercise 2 of homework 4 in [13]. Prim's algorithm for computing a minimum spanning tree is, e.g., described in Section 24.2 of [6]. 5: The Stable Marriage Problem Exercise 1 of homework 6 in [13]. Gale and Shapley were the rst to investigate the stable marriage problem and gave an O(n 2 ) solution in [8]. 1 Algorithms January 17, 2000 Homework 3 1: Minimum Spanning Trees A linear time algorithm for nding a minimum spanning tree for planar graph was rst given in [5]. The O(m log n) time algorithm for nding a minimum spanning tree in a general graph was described in [7]the paper introducing Fibonacci heaps. The current best dertministic minimum spanning tree algorithms use time O(m(m;n)), where is an inverse of Ackerman's function [4, 17]. A randomized
Literature Notes on Homeworks
, 2001
"... cise 2 of homework 4 in [13]. Prim's algorithm for computing a minimum spanning tree is, e.g., described in Section 24.2 of [6]. 5: The Stable Marriage Problem Exercise 1 of homework 6 in [13]. Gale and Shapley were the rst to investigate the stable marriage problem and gave an O(n 2 ) solution ..."
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cise 2 of homework 4 in [13]. Prim's algorithm for computing a minimum spanning tree is, e.g., described in Section 24.2 of [6]. 5: The Stable Marriage Problem Exercise 1 of homework 6 in [13]. Gale and Shapley were the rst to investigate the stable marriage problem and gave an O(n 2 ) solution in [8]. 1 Algorithms February 13, 2001 Homework 3 1: Minimum Spanning Trees A linear time algorithm for nding a minimum spanning tree for planar graph was rst given in [5]. The O(m log n) time algorithm for nding a minimum spanning tree in a general graph was described in [7]the paper introducing Fibonacci heaps. The current best dertministic minimum spanning tree algorithms use time O(m(m;n)), where is an inverse of Ackerman's function [4, 17]. A randomized linear time algorit