Time-Variant Spatial Network Model Vijay Gandhi, Betsy George (Group : G04) Group Project Overview of Database Research Fall 2006

Outline Motivation Related Work Problem Statement Contributions Future Work

Motivation A network is a system that allows its abstract representation as a graph Most networks change with time

Problem Definition Given : A set of frequent queries posed on a Network; Changes in the network with time. Output : A model that supports efficient and correct algorithms for computing the query results. Objective : Minimize storage and computation costs. Constraints : Edge cost are positive.

Related Work: Time Expanded Graphs N1 N2 N3 N N1 N2 N3 N N1 N2 N3 N t = 1 t = 3t = 2 N1 N2 N3 N4 Time Expanded Graph Representation Graph changing over time

A Conceptual Model t = 1 N1 N2 N3 N N1 N2 N3 N N1 N2 N3 N t = 3t = 2 Time series of snapshots

A Physical Model Time Aggregated Graph 1,1, 2,2,3 N1 N2 N3 N4 1,,4 2,2,2 8 8 Attributes are aggregated over edges and nodes. N1 N2 N3 N N1 N2 N3 N N1 N2 N3 N t = 3t = 2

Physical Model - Finding the shortest path Challenge Not all shortest paths display optimal substructure. Lemma 1: If there is a path from source to destination in the time aggregated graph then there is at least one optimal path which satisfies the optimal substructure property. How do we address this? [1,4] N2 N3 N4N1N5

Physical Model - Finding the shortest path Uses a greedy algorithm to find the shortest path At every step, Picks the node with the least cost. Chooses the edge that is available at the earliest.

Time Dependency of Shortest Paths Start time Shortest PathTravel time t=1N1-N2-N36 t=2N1-N2-N35 t=3N1-N2-N34 Shortest Path Travel time is time-dependent Best-Start-Time Shortest Path Shortest Paths N2 N1 N3 1,2,2,2,2,22,--,--,--,2,2

Best Start Time Shortest Paths Challenges Prefixes need not be optimal Optimal prefix can lead to longer waits N2 N1 N3 1,2,2,2,2,22,--,--,--,2,2 Start time Shortest Path Travel time t=1N1-N2-N36 t=3N1-N2-N34

Best Start Time Shortest Paths Strategies Minimize waits by postponing the start of the journey Reduce search space by forming path classes Journeys with consecutive start times Same travel times Pruning Node count in a shortest path is bounded by n. Eliminate irrelevant journeys

Experimental Design

Experimental Results

A Logical Model Examples of Logical Operators

Future Work Heuristics need to be explored for Best- start-time path algorithm to improve the performance. Add spatial properties to nodes, edges Extend spatial graph indexing methods to time-aggregated graphs Incorporate capacities on edges.

Questions?

Related Work Temporally Variable Transportation Networks (Ding,Guting, 2003) Models state changes over time ( ie. open, closed, blocked) Shortest paths computed at an instant Stochastic Networks (Miller-Hooks etal., 1998) Travel times are modeled as independent random variables. Computes least expected time paths. Computational complexity O(I 3 v 3 P) Graphs over time (Leskovec etal, 2005) Deals with how graphs evolve over time. Densification, Shrinking diameters

Related Work (contd.) Using flow networks (Fleisher etal, 2002) Uses time expanded networks Dynamic Graphs (Demetrsecu etal, 2004) Fully dynamic Algorithms that minimize recomputation.

Algorithm – Finding Shortest Path Initialize c[s] = 0;  v (  s), c[v] = ∞. Insert s in the priority queue Q. while Q is not empty do u = extract_min(Q); close u (C = C  {u}) for each node v adjacent to u do { t = min_t((u,v), c[u]); // min_t finds the earliest departure time for (u,v) If t +  u,v (t) < c[v] c[v] = t +  u,v (t) parent[v] = u insert v in Q if it is not in Q; } Update Q.

Cost Model For each node extracted from Q, for every adjacent node v, there is one edge schedule look up and a heap update.  O(log f E + log n). n : Number of nodes in the time aggregated graph f E : Frequency of edge presence Total cost = O (  [degree(v).(log f E + log n)]) = O(m(log f E + log n)) V  N

Patient Shortest Paths Path Classes P = (R, I(S,  )) R = e 1, …., e k S = t 1, ….., t k  - Validity interval uvw 34 13

Patient Shortest Paths Lemma : The number of maximal relevant path classes from u to v that contains k nodes is bounded by 2m f E Proof Sketch : P = (R, I(S,  )).  max – largest integer such that (R, I(S,  )) is valid. a)  =  max : t i +  is the upper extremity of presence interval of e i. b)  max >  : t i is the lower limit of presence interval of e i

Patient Shortest Path : Algorithm Key Components Append : Adds an edge (u,v) to the list of classes (P, I(S,  )) of paths from s to u. Merge : Merges the output of ‘append’ to existing list, while eliminating irrelevant paths.

Patient Shortest Path Algorithm Append : dep(u,v) – earliest availability of edge (u,v) after arrival(P,S). t u – upper limit of presence interval Case 1) arr(P, S) +   dep(u,v) Case 2) arr(P, S)  dep(u,v) uv dep(u,v) Arr(P,S)+  Class collapses to single path Advance entire schedule by  Set  = 0 Arr(P,S) dep(u,v) Advance entire schedule by  incr Set  =  -  incr  incr

Patient Shortest Path - Algorithm Case 2a) arr(P, S) +  > t u  incr tutu arr(P, S) +  Move schedule backward by  incr Add the modified schedule in the list of classes for ‘u’. Set  =  incr Merge merges the generated list with existing list eliminates irrelevant paths t dep1  t dep2 and t arr1  t arr2 P2 is irrelevant

Cost Model of Algorithm For each edge added, append costs O(mf E ) merge costs O(mf E ) For each node, append & merge cost = out_degree(u) * O(mf E ) For every increase in the length of path Cost = O(m 2 f E ) Total cost = O(nm 2 f E ) (because max. length of a path = n)