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Answering Distance Queries in directed graphs using fast matrix multiplication. Seminar in Algorithms Prof. Haim Kaplan Lecture by Lior Eldar 1/07/2007. Structure of Lecture. Introduction & History Alg1 – APSP Alg2 – preprocess & query Alg3 – Hybrid Summary. Problem Definition.

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answering distance queries in directed graphs using fast matrix multiplication

Answering Distance Queries in directed graphs using fast matrix multiplication

Seminar in Algorithms

Prof. Haim Kaplan

Lecture by Lior Eldar

1/07/2007

structure of lecture
Structure of Lecture
  • Introduction & History
  • Alg1 – APSP
  • Alg2 – preprocess & query
  • Alg3 – Hybrid
  • Summary
problem definition
Problem Definition
  • Given a weighted directed graph, we are requested to find:
    • APSP - All pairs shortest paths – find for any pair
    • SSSP - Single Source shortest paths – find all distances from s.
  • A hybrid problem comes to mind:
    • Preprocess the graph faster than APSP
    • Answer ANY two-node distance query faster than SSSP.
    • What’s it good for?
previously known results apsp
Previously known results – APSP
  • Undirected graphs
    • Approximated algorithm by Thorup and Zwick:
      • Preprocess undirected weighted graph in

expected time.

      • Generate data structure of size
      • Answer any query in O(1)
      • BUT: answer is approximate with a factor of 2k-1.
    • For non-negative integer weights at most M – Shoshan and Zwick developed an algorithm of run time
  • Directed graphs – Zwick - runs in
previously known results sssp
Previously known results - SSSP
  • Positive weights:
    • Directed graphs with positive weights – Dijkstra with
    • Undirected graphs with positive integer edge weights – Thorup with
  • Negative weights – much harder:
    • Bellman-Ford
    • Goldberg and Tarjan – assumes edge weight values are at least – N.
new algorithm by yuster zwick
New Algorithm by Yuster / Zwick
  • Solves the hybrid pre-processing-query problem for:
    • Directed graphs
    • Integer weights from –M to M
  • Achieves the following performance:
    • Pre-processing
    • Query answering – O(n)
  • Faster than previously known APSP (Zwick) so long as the number of queries is
  • Better than SSSP performance (Goldberg&Tarjan) for dense graphs with small alphabet – gap of
beyond the numbers
Beyond the numbers…
  • An extension of this algorithm allows complete freedom in optimization of the pre-processing - query problem.
  • to optimize an algorithm for an arbitrary number of queries q, we want: preprocessing time + q * query time to be minimal.
  • This defines the ratio between query time and pre-processing time - completely controlled by the algorithm inputs.
  • Meaning: if we know in advance the number of queries we can fine-tune the algorithm as we wish.
before we begin scope
Before we begin - scope
  • Assumptions:
    • No negative cycles
  • Inputs:
    • Directed Weighted Graph G=(V,E,w)
    • Weights are –M,…0,…,M
  • Outputs:
    • Data structure such that – given any two nodes – produces the shortest distance between them (and not the path itself) – with high probability.
matrix multiplication
Matrix Multiplication
  • The matrix product C=AB, where A is an matrix, B is , and C is matrix, is defined as follows:
  • Define: the minimal number of algebraic operations for computing the matrix product.
  • Define as the smallest exponent such that
  • Theorem by Coppersmith and Winograd:
distance products
Distance Products
  • The distance product , where A is an matrix, B is , and C is matrix, is defined as follows:
  • Recall: if W is an n x n matrix of the edge weights of a graph then is the distance matrix of the graph.
  • Lemma by Alon: can be computed almost as fast as “regular” matrix multiplication:
state of the art apsp
State-of-the-art APSP
  • Randomized algorithm by Zwick that runs in time
  • Intuition:
    • Computation of all short paths is intensive.
    • BUT: long paths are made up of short paths: once we pay the initial price we can leverage this work to compute longer paths with less effort.
  • Strategy: Giving up on certainty - with a small number of distance updates we can be almost sure that any long-enough path has at least one representative that is updated.
basic operations
Basic Operations
  • Truncation
    • Replace any entry larger than t with
  • Selection
    • Extract from D the elements whose row indices are in A, and column indices are in B.
  • Min-Assignment
    • Assign to each element the smallest between the two corresponding elements of D and D‘.
pseudo code
Pseudo-code
  • Simply sample nodes and multiply decimated matrices…
on matrices and nodes
On matrices and nodes…
  • Column-decimated matrix

Distance between any two nodes

D

Shortest directed path from any node to any node in B

on matrices and nodes 2
On matrices and nodes…(2)
  • Row-decimated matrix

Distance between any two nodes

Shortest directed path from any node in B to any node

what do we prove
What do we prove?
  • Lemma: if there is a shortest path between nodes i and j in G that uses at most edges, then after the -th iteration of the algorithm, with high probability we have
  • Meaning: at each iteration we update with high probability all the paths in the graph of a certain length. This serves as a basis for the next iteration.
proof outline
Proof Outline
  • By Induction:
    • Base case: easy – the input W contains all paths of length
    • Induction step:
      • Suppose that the claim holds for and show that it also holds for
      • Take any two nodes that their shortest distance is at least . The -th iteration matrix product will (almost certainly) plug in their shortest distance at location (i,j) of D.
slide18

i

k

j’

j

i’

Why?
  • Set
  • The path p from i to j is at least 2s/3.
  • This divides p into three subsections:
    • Left – at most s/3
    • Right – at most s/3
    • Middle – exactly s/3
the details
The Details
  • The left and right “thirds” - help attain the induction step.
    • The path p(i,k) and p(k,j) are short enough – at most 2s/3  good for previous step:
  • The middle “third” – ensures the fault probability is low enough.
    • Prob(no k is selected) =
    • Probability still goes to 0 (as n tends to infinity) after computation of
      • entries
      • iterations
slide20
So…
  • Assuming all previous steps were good enough:
    • With high probability each long-enough path has a representative in B
    • The update of the D using the product

plugs in the correct result.

  • Note that:
    • Each element is first limited to s*M
    • This is necessary for the fast-matrix-multiplication algorithm
complexity
Complexity
  • Where does the trick hide?
    • The matrix alphabet increases linearly with iteration number
    • The product size decreases with iteration number
  • For each iteration :
    • Alphabet size: s*M
    • Product complexity: , where
    • Total:
  • Disregarding the log function, and optimizing between fast and naïve matrix products we get:
fast product versus naive
Fast Product versus Naive

*assuming small M

complexity behavior
Complexity Behavior
  • For a given matrix alphabet M, we find the cross-over point between the matrix algorithms.
  • For high r (>M-dependent threshold) we use FMM
    • Complexity dependent on M
  • For low r (<threshold) we use naïve multiplication
    • Complexity not dependent on M
  • Q: How does complexity change over the iteration number?
pre processing algorithm
Pre-processing algorithm
  • Motivation:
    • We rarely query all node-pairs
  • Strategy:
    • Replace the costly matrix product

with 2 smaller products:

    • Generate data structure such that each query costs only
starting with the query
Starting with the query…
  • Pseudo-code:
  • What is a sufficient trait of D, such that the returned value will be, with high probability
  • Answer: with high probability, a node k on the path from i to j should have:
new matrix type
New matrix type
  • Row&Column-decimated matrix

Query data structure for any two nodes

D

Query data-structure for any 2 nodes in B

what do we prove1
What do we prove?
  • Lemma 4.1: If or , and there is a shortest path from i to j in G that uses at most edges, then after the -th iteration of the preprocessing algorithm, with high probability we have .
  • Meaning: D has the necessary trait: for any path p, if we iterate long enough, then with high probability, for at least one node k (in p(i,j)) the entries d(i,k), d(k,j) will contain shortest paths. Hence, “query” will return the correct result.
proof outline preprocess
Proof Outline - preprocess
  • By Induction:
    • Base case: easy – B=V, and the input W contains all paths of length .
    • Induction step:
      • Suppose that the claim holds for and show that it also holds for
      • Take any two nodes that their shortest distance is at most . The l-th iteration matrix products (2) will (almost certainly) plug in their shortest distance at location (i,j) of D provided that EITHER or

.

slide30

i

k

j’

j

i’

Why?
  • Set
  • The path p from i to j is at least 2s/3.
  • This divides p into three subsections:
    • Left – at most s/3
    • Right – at most s/3
    • Middle – exactly s/3
the details1
The Details
  • Assume that .
  • With high probability ( ) there will be k in p(i,j), such that (remember why?)
  • Both are also in ,since
  • We therefore attain the induction step:
    • The path p(i,k) and p(k,j) are short enough – at most 2s/3  good for previous step.
    • The end-points of these paths (k) are in
    • Therefore their shortest distance is in D
    • The second product then updates correctly. (assumption critical here)
where s the catch
Where’s the catch?
  • In APSP, we assure that:
    • At every iteration l we compute the shortest path of length at most .
    • BUT: we had to update all pairs each time
  • In the preprocess algorithm, we assure:
    • At every iteration l, we compute the shortest path of length at most only for a selected subset.
    • BUT: this subset covers all possible subsequent queries, with high probability.
complexity1
Complexity
  • Matrix product: instead of operations we only get
  • As before, for each iteration , the alphabet size is s*M.
  • Total complexity:
  • No matrix-product switch here!
performance
Performance
  • For small M, as long as the number of queries is less than we get better results than APSP.
  • For small M:
    • The algorithm overtakes Goldberg’s algorithm, if the graph is dense
    • For a dense-enough graph , we can run many SSSP queries and still be faster:
the larger picture
The larger picture
  • We saw:
    • Alg1: heavy pre-processing, light query
    • Alg2: light pre-processing, heavy query
    • Alg3: ?

Query-oriented (APSP)

Preprocess- oriented (pre-process)

the third way
The Third Way
  • Suppose we know in advance the we require no more than queries.
  • We use the following:
    • Perform iterations of the APSP algorithm
    • Perform iterations of the pre-process algorithm
    • Take the matrix B from the last step of step 1. The product returns in any shortest-distance query.
slide37
Huh?
  • After the first stage  D holds all the shortest path of all “short” paths, of lengths at most with high probability.
  • When the second starts stage it can be sure that the induction holds for all
  • The second stage takes care of the “long” paths, with respect to querying. Meaning:
    • If the path is long it will have a representative in one of the second-phase iterations
    • If it is too-short – it will fall under the jurisdiction of the first stage.
complexity2
Complexity
  • The first stage ( updates) costs at most
  • The second stage costs only
  • The query costs
  • For example – if want to answer a distance query in , we can pre-process in time
q a i ask you answer
Q&A (I ask - you answer)
  • Q: Why couldn’t we sample B in the query step of Alg2 – the one that initially costs O(n)?
  • A: Because if the path is too short – we will have no guarantee that it will have a representative in B. Alg3 solves this because short distances are computed rigorously.
  • Conclusion: the less we sample out of V when we query, the more steps we need to run APSP to begin with.
final procedure
Final Procedure
  • Given q queries, determine the query complexity using .
  • This assumes M is small enough so that we use fast product. Otherwise compare to
  • Execute alg3 using steps of APSP and steps of pre-process
  • Query all q queries.
summary
Summary
  • For the problem we defined: directed graph, with integer weights, whose absolute value is at most M, we have seen:
    • Alg1: State-of-the-art APSP in
    • Alg2: State-of-the-art SSSP in
    • Alg3: A method to calibrate between the two, for a known number of queries.