i/o-efficient graph algorithms

I/O-Efficient Graph Algorithms

Norbert ZehDuke University

EEF Summer School on Massive Data SetsÅrhus, Denmark

June 26 – July 1, 2002

Motivation

For theoreticians:• Graph problems are neat, often difficult, hence

interesting

For practitioners:• Massive graphs arise in GIS, web modelling, ...• Problems in computational geometry can be

expressed as graph problems• Many abstract problems best viewed as graph

problems• Extreme: Pointer-based data structures =

graphs with extra information at their nodes

Outline

Fundamental graph problems• List ranking• Algorithms for trees

• Euler tour• Tree labelling

• Graph searching• BFS/DFS

• Connectivity• Connected components• Minimum spanning tree

• Single source shortest paths

Outline

• Techniques and data structures• Graph contraction• Time-forward processing• Tournament tree• Buffered repository tree

• Lower bounds• List ranking• Connectivity

• Planar graphs

Introduction and “Simple” Problems

• List ranking• Euler tour• Tree labelling• Evaluating directed acyclic graphs• Greedy graph algorithms

List Ranking

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Why Is List Ranking Non-Trivial?

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The internal memory algorithm spends (N) I/Os in the worst case.

An Efficient List Ranking Algorithm

• Assume an independent set of size at least N/3 can be found efficiently (in O(sort(N)) I/Os).

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An Efficient List Ranking Algorithm

• Compressing L:• Sort elements in L \ I• Sort elements in I by their successor

pointers• Scan the two lists to update the label of

succ(v), for every element v I

• The I/O-complexity of this procedure is

Theorem: A list of size N can be ranked in O(sort(N)) I/Os.

NsortONsortOINI 3N2

The Euler Tour Technique

Goal: Given a tree T, represent it by a list L so that certain computations on T can be performed by ranking L.

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The Euler Tour Technique

Theorem: Given the adjacency lists of the vertices in T, an Euler tour can be constructed in O(scan(N)) I/Os.

• Let {v,w1},…,{v,wr} be the edgesincident to v

• Then succ((wi,v)) = (v,wi+1)) v

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Rooting a Tree

• Choosing a vertex r as the root of a tree T defines parent-child relationships between adjacent nodes

• Rooting tree T =computing for every edge{v,w} who is the parentand who is the child

• v = p(w) if and only ifrank((v,w)) < rank((w,v))

Theorem: A tree can be rooted in O(sort(N)) I/Os.

Computing a Preorder Numbering

Theorem: A preorder numbering of a rooted tree T can be computed in O(sort(N)) I/Os.

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preorder#(v) = rank((p(v),v))

Computing Subtree Sizes

Theorem: The nodes of T can be labelled with their subtree sizes in O(sort(N)) I/Os.

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Evaluating a Directed Acyclic Graph

• More general: Given a labelling , compute a labelling so that (v) is computed from (v) and (u1),…,(ur), where u1,…,ur are v’s in-neighbors

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Time-Forward Processing

• Assume nodes are given in topologically sorted order.

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Use priority queue Q to send data along the edges.

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Time-Forward Processing

Analysis:• Vertex set + adjacency lists scanned O(scan(|V| + |E|)) I/Os• Priority queue:

• Every edge inserted into and deleted from Qexactly once

O(|E|) priority queue operations O(sort(|E|)) I/Os

Time-Forward Processing

Analysis:• Vertex set + adjacency lists scanned O(scan(|V| + |E|)) I/Os• Priority queue:

• Every edge inserted into and deleted from Qexactly once

O(|E|) priority queue operations O(sort(|E|)) I/Os

Theorem: A directed acyclic graph G = (V,E) can be evaluated in O(sort(|V| + |E|)) I/Os.

Maximal Independent Set (MIS)

Algorithm GREEDYMIS:1. I 02. for every vertex v G do3. if no neighbor of v is in I then4. Add v to I5. end if6. end for

Maximal Independent Set (MIS)

Algorithm GREEDYMIS:1. I 02. for every vertex v G do3. if no neighbor of v is in I then4. Add v to I5. end if6. end for

Observation: It suffices to consider all neighbors of v which have been visited in a previous iteration.

Maximal Independent Set (MIS)

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Maximal Independent Set (MIS)

Theorem: A maximal independent set of a graphG = (V,E) can be computed in O(sort(|V|+|E|)) I/Os.

Large Independent Set of a List

Corollary: An independent set of size at least N/3 for a list L of size N can be found in O(sort(N)) I/Os.

• Every vertex in an MIS I prevents two other vertices from being in I:

Every MIS has size at least N/3.

Graph Connectivity

• Connected components• Minimum spanning tree

ConnectivityA Semi-External Algorithm

ConnectivityA Semi-External Algorithm

Analysis:• Scan vertex set to load vertices into main

memory• Scan edge set to carry out algorithm• O(scan(|V| + |E|)) I/Os

Theorem: The connected components of a graph can be computed in O(scan(|V| + |E|)) I/Os, provided that |V| M.

ConnectivityThe General Case

Idea:• If |V| M

• Use semi-external algorithm• If |V| > M

• Identify simple connected subgraphs of G• Contract these subgraphs to obtain graph

G’ = (V’,E’) with |V’| c|V|, c < 1• Recursively compute connected

components of G’• Obtain labelling of connected components

of G from labelling of components of G’

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ConnectivityThe General Case

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ConnectivityThe General Case

Main steps:• Find smallest neighbors (easy)• Compute connected components of graph

H induced by selected edges• Contract each component into a single vertex

(easy)• Call the procedure recursively• Copy label of every vertex v G’ to all vertices

in G represented by v (easy)

ConnectivityThe General Case

• Every connected component of H has size at least 2 |V’| |V|/2 recursive calls

Theorem: The connected components of a graphG = (V,E) can be computed in I/Os. logEsortO M

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MVlog

ConnectivityThe General Case

• Later: BFS in O(|V| + sort(|E|)) I/Os Can be used to identify connected components

• When |V| = |E|/B, algorithm takes O(sort(|E|)) I/Os

• Can stop recursion after recursive calls

Theorem: The connected components of a graphG = (V,E) can be computed in I/Os.

EBVlog

EBVlogEsortO

Minimum Spanning Tree (MST)

Observation: Connectivity algorithm can be augmented to produce a spanning tree of G.

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Minimum Spanning Tree (MST)

To obtain a minimum spanning tree:• Choose edge of minimum weight incident to v

• Some book-keeping:• The weight of an edge e in the compressed

graph = the min weight of all edges represented by e

• When “e is added” to T, add in fact this minimum edge

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Minimum Spanning Tree (MST)

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Theorem: A MST of a graph G = (V,E) can be computed in I/Os. logEsortO M

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A Fast MST Algorithm

• Idea:• Assume MST can be computed in

O(|V| + sort(|E|)) I/Os• Again recursion can be stopped after

iterations

• Prim’s algorithm:

EBVlog

A Fast MST Algorithm

• Maintain superset of blue edges in priority queue Q

• When edge {v,w} of minimum weight is retrieved, test whether v,w are both in T• Yes discard edge• No Add edge to MST and add all edges

incident to w to Q, except {v,w}(assuming that w T)

Problem: How to testwhether v,w T.

A Fast MST Algorithm

• If v,w T, but {v,w} T, then both v and w have inserted edge {v,w} into Q

There are two copies of {v,w} in Q• They are consecutive Perform two DELETEMIN operations

• If {v,w} = {y,z}, discard both• Otherwise, add {v,w} to T and re-insert {y,z}

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A Fast MST Algorithm

Analysis:• O(|V| + scan(|E|)) I/Os for retrieving adjacency

lists• O(sort(|E|)) I/Os for priority queue operations

Theorem: A MST of a graph G = (V,E) can be found in O(|V| + sort(|E|)) I/Os.

Corollary: A MST of a graph G = (V,E) can be found in I/Os.

EBVlogEsortO

Graph Contraction and Sparse Graphs

• A graph G = (V,E) is sparse if for any graph H obtainable from G through a series of edge contractions, |E(H)| = O(|V(H)|).

• For a sparse graph, the number of vertices and edges in G reduces by a constant factor in each iteration of the connectivity and MST algorithms.

Theorem: The connected components or a MST of a sparse graph with N vertices can be computed in O(sort(N)) I/Os.

Three Techniques for Graph Algorithms

• Time-forward processing:• Express graph problems as evaluation

problems of DAGs• Graph contraction:

• Reduce the size of G while maintaining the properties of interest

• Solve problem recursively on compressed graph

• Construct solution for G from solution for compressed graph

• Bootstrapping:• Switch to generally less efficient algorithm as

soon as (part of the) input is small enough