discrete math 2 shortest paths using matrix

Discrete Math 2Discrete Math 2Shortest Paths Using MatrixShortest Paths Using Matrix

CIS112CIS112February 14, 2007February 14, 2007

2007 Kutztown University 2

OverviewOverview Previously:Previously:

In weighted graph . .In weighted graph . . Shortest path from #7 to all othersShortest path from #7 to all others Search matrix methodSearch matrix method

Now:Now: Problem 8.6.2Problem 8.6.2 Implement Floyd’s AlgorithmImplement Floyd’s Algorithm


StrategyStrategy 3 nested loops 3 nested loops

i i ≔≔ 1 to 6 1 to 6 j j ≔≔ 1 to 6 1 to 6 k k ≔≔ 1 to 6 1 to 6

Basic ruleBasic rule If d(j,i) + d(i,k) < d(j,k) . .If d(j,i) + d(i,k) < d(j,k) . . Then d(j,k) Then d(j,k) d(j,i) + d(i,k) d(j,i) + d(i,k)


Interpretation of MatrixInterpretation of Matrix Row designates Row designates from verticesfrom vertices Column designates Column designates to verticesto vertices SuppSuppose entry [2,3] is 7ose entry [2,3] is 7

From vertex #2 to vertex #3 . .From vertex #2 to vertex #3 . . cost of travel = 7cost of travel = 7


The Basic OperationThe Basic Operation

Seek – best cost route from Seek – best cost route from jj to to kk Initial entries = cost of direct route, Initial entries = cost of direct route,

i.e., cost of edge i.e., cost of edge jkjk No edge No edge cost = cost = ∞∞ Guarantees that any route found is Guarantees that any route found is

betterbetter



Old cost from Old cost from jj to to kk compared to . . compared to . . Cost of 2 step hopCost of 2 step hop

jj to to ii and and then then ii to to kk

If 2 step hop has better cost . .If 2 step hop has better cost . . Then it becomes the new Then it becomes the new jj to to kk cost cost



Initially old cost is edge costInitially old cost is edge cost Later . .Later . . Old cost is cost of best route found Old cost is cost of best route found

so far so far


DenotationsDenotations

Outer loop – the Outer loop – the i loopi loop Middle loop – the Middle loop – the j loopj loop Inner loop – the Inner loop – the k loopk loop


OperationOperation Loops work in tandemLoops work in tandem 2 outer loops {2 outer loops {ii & & jj loops} loops} 2 inner loops {2 inner loops {jj & & kk loops} loops} Proceed Proceed

» row by row &row by row &

» column by columncolumn by column

Update travel cost . . Update travel cost . . from vertex from vertex jj to vertex to vertex kk


Matrix for Weighted GraphMatrix for Weighted Graph

VtxVtx AA BB CC DD EE ZZ

AA ∞∞ 22 33 ∞∞ ∞∞ ∞∞

BB 22 ∞∞ ∞∞ 55 22 ∞∞

CC 33 ∞∞ ∞∞ ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


First Outer LoopFirst Outer Loop

i = 1i = 1 j = j = ≔≔ 1 to 6 1 to 6 k k ≔≔ 1 to 6 1 to 6


First Inner LoopFirst Inner Loop

i = 1i = 1 j = 1j = 1 k k ≔≔ 1 to 6 1 to 6


First Middle LoopFirst Middle Loop

d(1,1) = d(1,1) = ∞∞ d(1,1) + d(1,k) d(1,1) + d(1,k) ≮≮ d(1,k), d(1,k), ∀∀kk Therefore, no changeTherefore, no change


Second Middle LoopSecond Middle Loop

i = 1i = 1 j = 2j = 2 k k ≔≔ 1 to 6 1 to 6



k = 1k = 1 d(1,1) = d(1,1) = ∞∞ d(2,1) + d(1,1) = d(2,1) + d(1,1) = ∞∞ d(2,1) + d(1,1) = d(2,1) + d(1,1) = ∞∞ ≮≮ d(2,1) d(2,1) Therefore, no changeTherefore, no change


i=1; j=2; k = 2 i=1; j=2; k = 2 (2,1) + (1,2) < (2,2) (2,1) + (1,2) < (2,2) ≡ 4 < ∞≡ 4 < ∞


AA ∞∞ 22 33 ∞∞ ∞∞ ∞∞

BB 22 44 ∞∞ 55 22 ∞∞

CC 33 ∞∞ ∞∞ ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


i=1; j=2; k = 3 i=1; j=2; k = 3 (2,1) + (1, 3) < (2,3) (2,1) + (1, 3) < (2,3) ≡ ≡ 5 < 5 < ∞∞


AA ∞∞ 22 33 ∞∞ ∞∞ ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 ∞∞ ∞∞ ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


i=1; j=2; k i=1; j=2; k ≔≔ 4,5,6 4,5,6(2,1) + (1,k) < (2,k) (2,1) + (1,k) < (2,k) ≡ ∞ ≮ ≡ ∞ ≮ xx


AA ∞∞ 22 33 ∞∞ ∞∞ ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 ∞∞ ∞∞ ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


Third Middle LoopThird Middle Loop

i = 1i = 1 j = 3j = 3 k k ≔≔ 1 to 6 1 to 6


Third Middle LoopThird Middle Loop

k = 1k = 1 d(1,1) = d(1,1) = ∞∞ d(3,1) + d(1,1) = d(3,1) + d(1,1) = ∞∞ d(3,1) + d(1,1) = d(3,1) + d(1,1) = ∞∞ ≮≮ d(3,1) d(3,1) Therefore, no changeTherefore, no change


i=1; j=3; k = 2 i=1; j=3; k = 2 (3,1) + (1, 2) < (3,2) (3,1) + (1, 2) < (3,2) ≡ ≡ 5 < 5 < ∞∞


AA ∞∞ 22 33 ∞∞ ∞∞ ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 ∞∞ ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


i=1; j=3; k = 3 i=1; j=3; k = 3 (3,1) + (1, 3) < (3,3) (3,1) + (1, 3) < (3,3) ≡ ≡ 6 < 6 < ∞∞


AA ∞∞ 22 33 ∞∞ ∞∞ ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


i=1; j=3; k i=1; j=3; k ≔≔ 4,5,6 4,5,6 (3,1) + (1, k) < (3,k) (3,1) + (1, k) < (3,k) ≡ ≡ ∞ ≮ ∞ ≮ xx


AA ∞∞ 22 33 ∞∞ ∞∞ ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


44thth, 5, 5thth & 6 & 6thth Middle Loops Middle Loops

i = 1i = 1 j j ≔≔ 4 to 6 4 to 6 k k ≔≔ 1 to 6 1 to 6


44thth, 5, 5thth & 6 & 6thth Middle Loops Middle Loops

j j ≔≔ 4 to 6 4 to 6 d(j,1) = d(j,1) = ∞∞ d(j,1) + d(i,k) = d(j,1) + d(i,k) = ∞∞ d(j,1) + d(i,1) = d(j,1) + d(i,1) = ∞∞ ≮≮ d(j,k) d(j,k) Therefore, no changeTherefore, no change


Second Outer LoopSecond Outer Loop

i = 2i = 2 j = j = ≔≔ 1 to 6 1 to 6 k k ≔≔ 1 to 6 1 to 6


i=2; j=1; k = 1 i=2; j=1; k = 1 (1,2) + (2, 1) < (1,1) (1,2) + (2, 1) < (1,1) ≡ ≡ 4 < 4 < ∞∞


AA 44 22 33 ∞∞ ∞∞ ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


i=2; j=1; k = 2 i=2; j=1; k = 2 (1,2) + (2, 2) < (1,2) (1,2) + (2, 2) < (1,2) ≡ ≡ 6 6 ≮≮ 2 2


AA 44 22 33 ∞∞ ∞∞ ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


i=2; j=1; k = 3 i=2; j=1; k = 3 (1,2) + (2, 3) < (1,3) (1,2) + (2, 3) < (1,3) ≡ ≡ 7 7 ≮≮ 3 3


AA 44 22 33 ∞∞ ∞∞ ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


i=2; j=1; k = 4 i=2; j=1; k = 4 (1,2) + (2, 4) < (1,4) (1,2) + (2, 4) < (1,4) ≡ ≡ 7 7 < ∞< ∞


AA 44 22 33 77 ∞∞ ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


i=2; j=1; k = 5 i=2; j=1; k = 5 (1,2) + (2, 5) < (1,5) (1,2) + (2, 5) < (1,5) ≡ ≡ 4 4 < ∞< ∞


AA 44 22 33 77 44 ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞


i=2; j=1; k = 6 i=2; j=1; k = 6 (1,2) + (2, 6) < (1,6) (1,2) + (2, 6) < (1,6) ≡ ≡ ∞ ≮ ∞∞ ≮ ∞


AA 44 22 33 77 44 ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞



i = 2i = 2 j = 2j = 2 k k ≔≔ 1 to 6 1 to 6


Second Middle Loop k Second Middle Loop k ≔≔ 1 to3 1 to3 ii jj kk 22 22 11

(2,2) + (2,1) < (2,1) ?(2,2) + (2,1) < (2,1) ? 4 + 2 4 + 2 ≮≮ 2 2

22 22 22 (2,2) + (2,2) < (2,2) ?(2,2) + (2,2) < (2,2) ? 4 + 4 4 + 4 ≮≮ 4 4

22 22 33 (2,2) + (2,3) < (2,3) ?(2,2) + (2,3) < (2,3) ? 4 + 5 4 + 5 ≮≮ 5 5


Second Middle Loop k Second Middle Loop k ≔≔ 4 to6 4 to6 ii jj kk 22 22 44

(2,2) + (2,4) < (2,4) ?(2,2) + (2,4) < (2,4) ? 4 + 5 4 + 5 ≮≮ 5 5

22 22 55 (2,2) + (2,5) < (2,5) ?(2,2) + (2,5) < (2,5) ? 4 + 2 4 + 2 ≮≮ 2 2

22 22 66 (2,2) + (2,6) < (2,6) ?(2,2) + (2,6) < (2,6) ? 4 + 4 + ∞∞ ≮≮ ∞∞



AA 44 22 33 77 44 ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 ∞∞ 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞

Matrix After 2Matrix After 2ndnd Middle Loop Middle Loop {no change}{no change}


Third Middle Loop k Third Middle Loop k ≔≔ 1 to3 1 to3 ii jj kk 22 33 11

(3,2) + (2,1) < (3,1) ?(3,2) + (2,1) < (3,1) ? 5 + 2 5 + 2 ≮≮ 3 3

22 33 22 (3,2) + (2,2) < (3,2) ?(3,2) + (2,2) < (3,2) ? 5 + 4 5 + 4 ≮≮ 5 5

22 33 33 (3,2) + (2,3) < (3,3) ?(3,2) + (2,3) < (3,3) ? 5 + 5 5 + 5 ≮≮ 6 6


Third Middle Loop k Third Middle Loop k ≔≔ 4 to6 4 to6 ii jj kk 22 33 44

(3,2) + (2,4) < (3,4) ?(3,2) + (2,4) < (3,4) ? 5 + 5 < 5 + 5 < ∞∞

22 33 55 (3,2) + (2,5) < (3,5) ?(3,2) + (2,5) < (3,5) ? 5 + 2 5 + 2 ≮≮ 5 5

22 33 66 (3,2) + (2,6) < (3,6) ?(3,2) + (2,6) < (3,6) ? 5 + 5 + ∞∞ ≮≮ ∞∞



AA 44 22 33 77 44 ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 1010 55 ∞∞

DD ∞∞ 55 ∞∞ ∞∞ 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞

Matrix After 3Matrix After 3rdrd Middle Loop Middle Loop {1 cell changed}{1 cell changed}


Fourth Middle Loop k Fourth Middle Loop k ≔≔ 1 to3 1 to3 ii jj kk 22 44 11

(4,2) + (2,1) < (4,1) ?(4,2) + (2,1) < (4,1) ? 5 + 2 < 5 + 2 < ∞∞

22 44 22 (4,2) + (2,2) < (4,2) ?(4,2) + (2,2) < (4,2) ? 5 + 4 5 + 4 ≮≮ 5 5

22 44 33 (4,2) + (2,3) < (4,3) ?(4,2) + (2,3) < (4,3) ? 5 + 5 < 5 + 5 < ∞∞


Fourth Middle Loop k Fourth Middle Loop k ≔≔ 4 to6 4 to6 ii jj kk 22 44 44

(4,2) + (2,4) < (4,4) ?(4,2) + (2,4) < (4,4) ? 5 + 5 < 5 + 5 < ∞∞

22 44 55 (4,2) + (2,5) < (4,5) ?(4,2) + (2,5) < (4,5) ? 5 + 2 5 + 2 ≮ ≮ 11

22 44 66 (4,2) + (2,6) < (4,6) ?(4,2) + (2,6) < (4,6) ? 5 + 5 + ∞∞ ≮≮ 2 2



AA 44 22 33 77 44 ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 1010 55 ∞∞

DD 77 55 1010 1010 11 22

EE ∞∞ 22 55 11 ∞∞ 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞

Matrix After 4Matrix After 4thth Middle Loop Middle Loop {3 cells changed}{3 cells changed}


Fifth Middle Loop k Fifth Middle Loop k ≔≔ 1 to3 1 to3 ii jj kk 22 55 11

(5,2) + (2,1) < (5,1) ?(5,2) + (2,1) < (5,1) ? 2 + 2 < 2 + 2 < ∞∞

22 55 22 (5,2) + (2,2) < (5,2) ?(5,2) + (2,2) < (5,2) ? 2 + 4 2 + 4 ≮≮ 2 2

22 55 33 (5,2) + (2,3) < (5,3) ?(5,2) + (2,3) < (5,3) ? 2 + 5 2 + 5 ≮≮ 5 5


Fifth Middle Loop k Fifth Middle Loop k ≔≔ 4 to6 4 to6 ii jj kk 22 55 44

(5,2) + (2,4) < (5,4) ?(5,2) + (2,4) < (5,4) ? 2 + 5 2 + 5 ≮≮ 1 1

22 55 55 (5,2) + (2,5) < (5,5) ?(5,2) + (2,5) < (5,5) ? 2 + 2 < 2 + 2 < ∞∞ 22 55 66 (5,2) + (2,6) < (5,6) ?(5,2) + (2,6) < (5,6) ? 2 + 2 + ∞∞ ≮≮ 4 4



AA 44 22 33 77 44 ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 1010 55 ∞∞

DD 77 55 1010 1010 11 22

EE 44 22 55 11 44 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞



Sixth Middle Loop k Sixth Middle Loop k ≔≔ 1 to3 1 to3 ii jj kk 22 66 11

(6,2) + (2,1) < (6,1) ?(6,2) + (2,1) < (6,1) ? ∞∞ + 2 + 2 ≮ ∞≮ ∞ 22 66 22 (6,2) + (2,2) < (6,2) ?(6,2) + (2,2) < (6,2) ? ∞∞ + 4 + 4 ≮ ∞≮ ∞

22 66 33 (6,2) + (2,3) < (6,3) ?(6,2) + (2,3) < (6,3) ? ∞∞ + 5 + 5 ≮ ∞≮ ∞


Sixth Middle Loop k Sixth Middle Loop k ≔≔ 4 to6 4 to6 ii jj kk 22 66 44

(6,2) + (2,4) < (6,4) ?(6,2) + (2,4) < (6,4) ? ∞∞ + 5 + 5 ≮ ∞≮ ∞

22 66 55 (6,2) + (2,5) < (6,5) ?(6,2) + (2,5) < (6,5) ? ∞∞ + 2 + 2 ≮ ∞≮ ∞ 22 66 66 (6,2) + (2,6) < (6,6) ?(6,2) + (2,6) < (6,6) ? ∞∞ + + ∞∞ ≮ ∞≮ ∞



AA 44 22 33 77 44 ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 1010 55 ∞∞

DD 77 55 1010 1010 11 22

EE 44 22 55 11 44 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞



Third Outer LoopThird Outer Loop

i = 3i = 3 j = j = ≔≔ 1 to 6 1 to 6 k k ≔≔ 1 to 6 1 to 6 Step by step details given hereStep by step details given here



AA 44 22 33 77 44 ∞∞

BB 22 44 55 55 22 ∞∞

CC 33 55 66 1010 55 ∞∞

DD 77 55 1010 1010 11 22

EE 44 22 55 11 44 44

ZZ ∞∞ ∞∞ ∞∞ 22 44 ∞∞

Matrix After 3Matrix After 3rdrd Outer Loop Outer Loop {0 cells changed}{0 cells changed}


Fourth Outer LoopFourth Outer Loop




AA 44 22 33 77 44 99

BB 22 44 55 55 22 77

CC 33 55 66 1010 55 1212

DD 77 55 1010 1010 11 22

EE 44 22 55 11 22 33

ZZ 99 77 1212 22 33 44

Matrix After 4Matrix After 4thth Outer Loop Outer Loop {10 cells changed}{10 cells changed}


Fifth Outer LoopFifth Outer Loop




AA 44 22 33 55 44 77

BB 22 44 55 33 22 55

CC 33 55 66 66 55 88

DD 55 33 66 22 11 22

EE 44 22 55 11 22 33

ZZ 77 55 88 22 33 44



Sixth Outer LoopSixth Outer Loop




AA 44 22 33 55 44 77

BB 22 44 55 33 22 55

CC 33 55 66 66 55 88

DD 55 33 66 22 11 22

EE 44 22 55 11 22 33

ZZ 77 55 88 22 33 44




AA -- 22 33 55 44 77

BB 22 -- 55 33 22 55

CC 33 55 -- 66 55 88

DD 55 33 66 -- 11 22

EE 44 22 55 11 -- 33

ZZ 77 55 88 22 33 --

Final MatrixFinal MatrixDiagonal values are removedDiagonal values are removed


Path Costs from APath Costs from A Given by row entriesGiven by row entries A A B :: 2 B :: 2 A A C :: 3 C :: 3 A A D :: 5 D :: 5 A A E :: 4 E :: 4 A A Z :: 7 Z :: 7


Path Costs from BPath Costs from B Given by row entriesGiven by row entries B B A :: 2 A :: 2 B B C :: 5 C :: 5 B B D :: 3 D :: 3 B B E :: 2 E :: 2 B B Z :: 5 Z :: 5


Path Costs from CPath Costs from C Given by row entriesGiven by row entries C C A :: 3 A :: 3 C C B :: 5 B :: 5 C C D :: 6 D :: 6 C C E :: 5 E :: 5 C C Z :: 8 Z :: 8


Path Costs from DPath Costs from D Given by row entriesGiven by row entries D D A :: 5 A :: 5 D D B :: 3 B :: 3 D D C :: 6 C :: 6 D D E :: 1 E :: 1 D D Z :: 2 Z :: 2


Path Costs from EPath Costs from E Given by row entriesGiven by row entries E E A :: 4 A :: 4 E E B :: 2 B :: 2 E E C :: 5 C :: 5 E E D :: 1 D :: 1 E E Z :: 3 Z :: 3


Path Costs from ZPath Costs from Z Given by row entriesGiven by row entries Z Z A :: 7 A :: 7 Z Z B :: 5 B :: 5 Z Z C :: 8 C :: 8 Z Z D :: 2 D :: 2 Z Z E :: 3 E :: 3


Shortest PathsShortest Paths

Floyd’s Algorithm gives the path costsFloyd’s Algorithm gives the path costs Finding the actual paths reuqires Finding the actual paths reuqires

additional workadditional work We will need two matricesWe will need two matrices

One to hold travel costsOne to hold travel costs Other to hold actual pathsOther to hold actual paths

discrete math 2 shortest paths using matrix

Documents