Title: CS622 Page 1
1Graph Theory
- A loop is an edge where both endpoints are the
same - Two edges are parallel if they have the same end
points. - A graph is simple if there is no loops or
parallel edges. - The degree of a node is the number of edges in
the graph that have the node as an endpoint.The
degrees of nodes in a graph are important graph
property . - A component of a graph is a maximal connected
subgraph. - A Tree is a connected, simple graph without
cycles. - Any tree with n nodes has n-1 edges.
- Trees are optimal network designs when links have
very high capacity or enormously expensive, and
there is no reliability constraints.
2Minimum Cost Network
- Problem Given the distances among 4 cities,
build the minimum road to connect them. Assume
1000/km to build the road. - Solution Choose the shorter distance links first
and avoid those links that connect nodes already
included. This is Kruskals MST Algorithm.
3Minimum Cost Network
- Select Charmes-Duval, Bregen-Charmes,
- Avoid Bregen-Duval since both them already
included. - Choose Anagon-Charmes.
- The results is a star topology.
4Topology
- A tree is a star if only one node has degree
greater than 1. - A tree is a chain if no node has degree greater
than 2. - A graph G is weighted if there is a real number
associated with each edge. It is denoted as
(G,w). - The weight of an edge e is denoted as w(e).
- The weighting function w E?R.
- If G is connected, we like to select a connected
subgraph with minimum weight.
5Kruskals MST Algorithm
- MST ? Minimum Spanning Tree.
- Check if graph is connected. If not, abort.
- Sort the edges in ascending order by weight.
- Mark each node as separate component.
- Loop on the edges until we accepted V-1
edges.Let e be the candidate edge. - If the endpoints are in different components,
merge the two components and accept the edge and
increment the of edges accepted by 1.
6Prims MST Algorithm
- Start with a node v, no edge, T0(v, ?).
- Add least expensive edge.
- 1 Tree Prim(Graph G, Node root)
- 2
- 3 for_each(node, G?nodes)
- 4 node? label INF
- 5 node?intree FALSE
- 6
- 7 root?label 0
- 8 root?pred root
- 9 NodesChosen 0
- 10
7- 11 while (Nodeschosen lt NumberNodes(G))
- 12 nodeFindMinLabel(G) / Only nodes
not in the tree are checked. / - 13 node?intree TRUE
- 14 NodeChosen
- 15 for_each(node2, node?adjacent_nodes)
- 16 if (node2?intree FALSE
- 17 node2?label gt
weight(edge(node, node2)) - 18 node2-gtprednode
- 19 node2?labelweight(edge(node,
node2)) - 20 / endif /
- 21 / endfor /
- 22 / endwhile /
- 23
- 24 treecreate_Tree()
- 25 BuildTreeFromPreds(G, tree)
- 26 return(tree)
- 27 / end Prim /
8Use Delite to Calculate MSTs
- Start Delite. Select Start Programs Delite.
- Select File Read Input File Prim.INP.
- The nodes are displayed.
- Select Design Prim for displaying the result of
the Prim MST Algorithm. - The legend window shows the link_cost of the MST.
- Select Display Select Map File USAVH.met for
overlay Map file. - The result display is shown next.
- Select Design Set Input Parameters Trace
Yes to generate .trc trace file. - Use Notepad/wordpad to view the Prim.trc or
Prim.INP file.
9(No Transcript)
10Tree Designs
- MST is good when
- Links are highly reliable
- Networks can tolerate low reliability
- The number of sites are small.
- Either Prims or Kruskals algorithm gives
optimal solution. - MSTs are not good networks to use when the
number of nodes is large.
11Squreworld Counter-example
- 1000 miles x 1000 miles.
- One type of transmission lines? 1Mbps.
- Cost for two sites with locations (x1, y1) and
(x2, y2) with distance d (100010d)/month. - Consider problems with 5, 10, 20, 50, and 100
sites, traffic are normalized with 1kbps from one
site to another. The traffic volumes grows
quadratically
12MST for 5 Node Network
13Generate Files for Squreworld Counter-example
- Compile the c\Program Files\Delite\Source\gen.c
- Run gen ltnumber of nodesgt ltfilename.gengtto
produce the 5, 20, 50, 100 nodes files. - Run delite, select File Generate Input
- Select the corresponding .gen file.
- The monitor will show the files are successfully
generated. They include .REQ (traffic
requirement), .CST (cost file.). - Edit the .REQ file, replace the link bandwidth
with 1000.
14MST for 10 Node Network
15Legginess? in Network
- Traffic in the above 10 node MST takes a
circuitous route between source and destination. - To quantify the legginess in the network, we
define - Definition 3.17 The number of hops (hop count)
between node n1 and n2 is the number of edges in
the path chosen by the routing algorithm for the
traffic flowing from n1 to n2. If only one path
is chosen or if all paths chosen have the same
number of edges, then we denote the number by
hops(n1, n2).
16Average of Hops
- Although the two routes between a pair of nodes
can be asymmetric, there is only one path between
a pair of nodes in a tree. - Definition 3.18 The average number of hops in a
network, - The average number of hops is quite important in
evaluating MST designs. - The sum of the traffic on all links Total
traffic x hops
17MST for 20 Node Network
18MST for 50 Node Network
19MST for 100 Node Network
20Traffic Volume and Costs
- The traffic of 100 node MST grows 5 orders of
magnitude from 20kbps of 5 node MST. - 1.5 order of magnitude comes only from average
hop count. - Problem MSTs tend to have very long and
circuitous paths.
21Shortest Path Trees
- Definition 3.19 Given a weighted graph (G, w)
and nodes n1 and n2, the shortest path from n1 to
n2 is a path such that ?e?P w(e) is a minimum. - Segments of a shortest path are the shortest
paths of their end points. (recall the optimality
principle) - Definition 3.20 Given a weighted graph (G, w)
and a node n1, a shortest path tree (SPT) rooted
at n1 is a tree s.t. for any other node n2 ? G,
the path from n1 to n2 in the tree T is a
shortest path between the nodes.
22SPT for 20 Node Network
Generate using Prim-Dijkstar with centerN14
alpha1.0
MST SPT Avg(HOPS)
5.0316 1.9000 Max_Util 0.010 0.002
23SPT vs. MST
- Queueing delay and Utilization
- Let
- Then
- Average packet delay
- 20 node networks
- Average packet delay for MSTCost48,686.
- Average packet delay for SPT Cost88,612.
higher cost, lower delay, N14 failure!
24Prim-Dijkstra Tree
- Try to find trees falls between MST and SPT.
- Both Prim and Dijkstra algorithm start with
initial label, looping over nodes to find one
with smallest label, bringing it into the tree,
finally relabeling all the neighbors. - Prim-Dijkstra Tree uses the following
label.Where 0 ???1 is used to parameterize
the algorithm. - ?0, we build MST. ?1, we build an SPT from
root.
25Choosing Prim-Dijkstra Trees
?0.3 and ?0.4 give attractive trees.
26Dominace among Designs
- Impose a partial ordering when picking the
designs. - Definition 3.21 Given a set S and an operator ?
that maps S x S ? TRUE, FALSE, then we call S
a partial ordered set, or poset, if - For any s ? S, s ? s is FALSE.
- For any s1,s2 ? S, s1 ? s2, if s1 ? s2 is True,
then s2 ? s1 is FALSE. - If s1 ? s2 and s2 ? s3 are TRUE, then s1 ? s3 is
TRUE. - Note that there may exist 2 elements s1,s2 ? S,
s.t. neither s1 ? s2 nor s2 ? s1 is TRUE. They
are called incomparable.
27Apply POSET to Pick Designs
- Definition 3.22 Suppose D1 has cost C1 and
performance P1. Suppose D2 has cost C2 and
performance P2. We will says D1 dominates D2, or
D1 ? D2, if C1 lt C2 and P1 gt P2. - Use the above definition, we compute the
domination partial order in Table 3.12.
28Domination Partial Ordering
- N0, N1, N2 are dominated by others. Remove them
from consideration list.
29Consider Marginal Cost of Delay
- There are still 6 designs to consider.
- The ratio (C1-C2)/(P2-P1) gives the cost for
delay. - 5099 buy 17 ms delay (from N3 to N4)the cost
for delay5099/0.017305329. - Between N5 and N6, the cost for delay rises 5
times.
30Tours
- Sometime trees are just too unreliable.
- There are designs that are far more reliable, yet
have only one additional link ? Tours. - It is a possible solution of the traveling
salesman problem (TSP). - Definition 3.24 Given a set of vertices v1, v2,
, vn, a tour T is a set of n edges E s.t. each
vertex v has degree 2 and the graph is connected. - The tour can be represented as a permutation
(vt1, vt2,, vtn). There are n! such
permutations. - Since cyclically permuting them and the reverse
permutation give the same tour , we have (n-1)!/2
tours.
31TSP Problem
- Definition 3.25 Given a set of vertices (v1, v2,
, vn) and a distance function d V x V?R, the
traveling salesman problem is to find the tour T
such that is a minimum. In this we identify
vt n1 with vt1. -
32Network Reliability
- Definition 3.26 The reliability of a network is
the probability that the functioning nodes are
connected by working links. - Assume the probability of each node working is 1
and the probability of a link falling is p.
Typically p is very small. Let q 1 p. - For a 5-node tree, Ptree(failure)1-(1-p)4?4p
- For a 5-node tour, Ptour(failure)1-((1-p)55p(1-p
)4)1-(q55pq4) 10p2q310p3q25p4qp5 - substitute with terms from 1(pq)5
33Ptree(failure) vs. Ptour(failure)
- When p10-6, the reliability difference is 5
orders of magnitude difference.
34Nearest Neighbor Algorithm
- Start at a root node. Set current_node root
node. - Loop until all nodes in the tour
- Go through the node list, find the node closest
to the current_node that is not in the tour. Let
this node be best_node. - Create an edge between current_node and
best_node. - Set current_node to be best_node.
- Finally create an edge between the last node and
root node.
35What is wrong with this tour?
36Uncrossing the Tour
37Creditable Algorithms
- Definition 3.27 A heuristic optimization
algorithm produces a creditable result if the
result is a local optimum for the problem.
Otherwise, it produces an uncreditable result. - We are asking absence of stupidity, not
performance. - Definition 3.28 A suite of network design
problems is a set of triples (Locationsi,
Traffici, Costsi) for i1,,S - Definition 3.29 A creditability test is a
program test (net, traffic, cost) that takes a
network problem as input and return OK or FAIL
depending on whether or not test() can manipulate
net into another valid network of lower costs. - Definition 3.30 Given a suite of network design
problems S, a design algorithm A, and a
creditability test t(),then
38Creditability of Tours built by Nearest-Neighbor
Algorithm
- Test program in appendix B, generate a large set
of locations. - Start at location 0, build a tour using N-N
algorithm. - Call test_tour() to apply line-crossing test
cross() O(n2) times. - Collect the statistics.
39A More Creditable N-N Heuristic
- Improvement 1 The closest node to the list of
node in the tour, instead of to the last node in
the list. - Improvement 2 The closest node can be added to
any place in the tour, not just append to the
end. The insert location minimize the following
formuladist(Ni, best_node)dist(best_node,Nj)
dist(Ni,Nj) - Intuitively, you try to find a place where Ni,
best_node, Nj is as flat as possible (follow
straight line). - O(2n2) complexity instead of O(n2) for basic N-N
algorithm.
40Result of MC-NN Algorithm
- Note that it performs much better.
- Instead selecting nearest neighbor, we can select
the furthest neighbors (they are found isolated
at the end of N-N algorithms and have poor choice
to join the tour.
41TSP Tour Do not Scale
- Theorem 3.3 Given uniform traffic any TSP tour
of n nodes has avg(hops)(n1)/4 if n is odd, and
n2/(4(n-1) if n is even. - At 100 node tour, for the avg hop count, TSP tour
is twice as bad as MST.
422-connectivity Graph
- Tour is a-connected graph. They survive the lost
of a node. - Definition 3.31 Given a connected graph G(V,E),
the vertex v is an articulation point if removing
the vertex and all the attached edges disconnects
the graph. - In tree, any node has degree gt 1 is a
articulation pt. - Definition 3.32 If a connected graph G(V,E) has
no articulation points, then the graph is
2-connected.
43Unobvious Articulation Point
Easy to detect
Not easy to detect
442-connectivity Theorem
- Theorem 3.4 Suppose G1(V1,E1) and G2(V2,E2)
are 2-connected graphs with V1 ?V2 ?. Let v1,
v2 ? V1 and v3, v4 ? V2. Then the graph G with
vertices V1 ? V2 and Edges E1 ? E2 ? (v1, v3) ?
(v2, v4) is 2-connected. - Divide and Conquer Divide the nodes into
clusters, find the tour for each cluster, treat a
cluster a node and connect clusters with a ring
of edges that do not share the same vertex. - Note that for n clusters, the connected tours
will have n more edges (more reliable) and
shorter average hop count.
45Original 20-node Tour
462 Cluster Tour
47Join the Tours
48Theorem 3.5
- Suppose that G(V,E) is a 2-connected graph with
V gt 2. Suppose that each node vi ? V is
replaced by a 2-connected graph Gi. Suppose each
edge e(u,w) ? E is replaced by an edge e from
u ? Gu to v ? Gw. Then if no 2 of these
replacement edges have a common vertex, the graph
H(?iVi, ?iEi?E) is a 2-connected graph.
494-cluster Design for 50 nodes
Single tour HOP12.755 4-cluster HOP6.8971
50Exercise 6
- Problem 1. Network Restoration.
- For link A-G cut, show the restoration paths
(route as sequence of nodes, and itsrestored
bandwidth. - Discuss the selection trade-offfor the two
algorithms at Page5 handout. - Problem 2. Tree and Tour.
- Problem 3.7.
- Explain why for the 20-node random problem
avg(hops)SPT1.9000. - Prove for a start on n nodes,S, that
avg(hops)S2-2/n. - Download the g20sw. from cs622/public_html/delit
e.Run delite with 1. Tour (Nearest Neighbor), 2.
Tour (Furthest Neighbor), 3. Prim, 4. SPT.
Compare the cost, avg(hops), and max. utilization
of these four designs.
51Solution to EX6
- Problem 1. Network Restoration
- For link A-G cut, show the restoration paths
(route as sequence of nodes, and its restored
bandwidth. - Ans 8 working channels lost.3 restoration paths
through AF.3 restoration paths through ACEF1
restoration path through ABDEF - b. Discuss the selection trade-offfor the two
algorithms at Page5 handout. - Ans Depends on the restoration deadline and
whether restoration paths are needed earlier for
the priority traffic.If the restoration level is
the utmost important metric, then the final
restorationlevel at the restoration deadline
decides theselection of the algorithm. The
curve however does not show the spare channel
usage.
52Solution to HW5 (2)
- Problem 2. Tree and Tour.
- Problem 3.7.
- Explain why for the 20-node random problem
avg(hops)SPT1.9000. - Prove for a star on n nodes,S, that
avg(hops)S2-2/n. - Ans 1 There are c(20,2)2019/2190
connections.Among them, 19 are to the
star.avg(hops)spt((190-19)2191)/1901.9 - Ans 2 There are c(n,2)n(n-1)/2
connections.Among them, n-1 are to the
star.Avg(hops)S(n(n-1)/2-(n-1))2(n-1))/(n(n-
1)/2)(n(n-1)-(n-1))/(n(n-1)/2)2(n-1)/n2-2/n
. - Download the g20sw. from cs622/public_html/delit
e.Run delite with 1. Tour (Nearest Neighbor), 2.
Tour (Furthest Neighbor), 3. Prim, 4. SPT.
Compare the cost, avg(hops), and max. utilization
of these four designs.
53Delite Tool Analysis (Tour NN)
54Tour (Furthest Neighbor)
55PRIM
56SPT
57Network Design Comparison