Graphs - profs.info.uaic.rosd/eng/res/eng_curs-07.pdf · DNA word design in biomolecular computing...

Graphs

DS 2019/2020

Content

Abstract data type Graph

Abstract data type Digraph

The implementation with adjacency matrices

The implementation with adjacency linked lists

Graph traversal algorithms (DFS, BFS)

Finding the (strongly) connected components

FII, UAIC Lecture 7 DS 2019/2020 2 / 51

Graphs

I G = (V ,E )I V a set of verticesI E a set of edges; an edge = a non-ordered pair of distinct vertices

V = {0, 1, 2, 3}E = {{0, 1}, {0, 2}, {1, 2}, {2, 3}}u = {0, 1} = {1, 0}

0,1 - the ends of uu is incident with 0 and 10 and 1 are adjacent (neighbors)


Graphs

I Walk from u to v : u = i0, {i0, i1}, i1, · · · , {ik−1, ik}, ik = v3, {3,2}, 2, {2,0}, 0, {0,1}, 1, {1,3},3, {3,2},2

I Trail: a walk where any two edges are distinct

I Circuit = a closed trail where any two intermediate edges are distinct

I Path: a walk where any two vertices are distinct

I Cycle: closed path (i0 = ik)


Induced subgraph

I G = (V ,E ) – a graph, W – a subset of V

I Induced subgraph: G ′(W ,E ′), whereE ′ = {{i , j}|{i , j} ∈ E si i ∈W , j ∈W }


Graphs - Connectivity

Any graph can be expressed as the disjoint union of induced, connectedand maximal subgraphs (connected components).

I i R j if and only if there is a path from i to j

I R is an equivalence relation

I V1, · · · ,Vp equivalence classes

I G1, · · · ,Gp connected components, where Gi = (Vi ,Ei ) a subgraphinduced by Vi

I connected graph = a graph with a single connected component



I objects:I graphs G = (V ,E ), V = {0, 1, · · · , n − 1}

I operations:I emptyGraph()

I input: nothingI output: the empty graph (∅, ∅)

I isEmptyGraph()I input: G = (V ,E),I output: true if G = (∅, ∅), false other way

I insertEdge()I input: G = (V ,E), i , j ∈ VI output: G = (V ,E ∪ {i , j})

I insertVertex()I input: G = (V ,E), V = {0, 1, · · · , n − 1}I output: G = (V ′,E), V ′ = {0, 1, · · · , n − 1, n}



I removeEdge()I input: G = (V ,E ), i , j ∈ VI output: G = (V ,E –{i , j})

I removeVertex()I input: G = (V ,E ), V = {0, 1, · · · , n − 1}, kI output: G = (V ′,E ′), V ′ = {0, 1, · · · , n − 2}

{i ′, j ′} ∈ E ′ ⇔ (∃{i , j} ∈ E ) i 6= k , j 6= k,

i ′ = if (i < k) then i else i − 1,

j ′ = if (j < k) then j else j − 1



I adjacencyList()I input: G = (V ,E ), i ∈ VI output: the list of vertices adjacent with i

I listOfReachableVertices()I input: G = (V ,E ), i ∈ VI output: the list of vertices reachable from i


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Digraph (directed graph)

I D = (V ,A)I V a set of verticesI A a set of arcs (directed edges); an arc = an ordered pair of distinct

vertices

V = {0, 1, 2, 3}A = {(0, 1), (2, 0), (1, 2), (3, 2)}a = (0, 1) 6= (1, 0)

0 – the tail of a1 – the head of a


Digraph

I Walk: i0, (i0, i1), i1, · · · , (ik−1, ik), ik3, (3,2), 2, (2,0), 0, (0,1), 1, (1,2), 2, (2,0), 0

I Trail: a walk where any two edges are distinct

I Circuit = a closed trail where any two intermediate edges are distinct

I Path: a walk where any two vertices are distinct

I Cycle: closed path (i0 = ik)


Digraph - Connectivity

I i R j if and only if there is a path from i to j and a path from j to i

I R is an equivalence relation

I V1, · · · ,Vp the equivalence classes

I G1, · · · ,Gp strongly connected components, where Gi = (Vi ,Ai ) thesubdigraph induced by Vi

I strongly connected digraph = digraph with a single stronglyconnected component

I connected digraph

V 1 = {0, 1, 2}A1 = {(0, 1), (1, 2), (2, 0)}

V 2 = {3}A2 = ∅



I objects: digraphs D = (V ,A)I operations:

I emptyDigraph()I input: nothingI output: the empty digraph (∅, ∅)

I isEmptyDigraph()I input: D = (V ,A),I output: true if D = (∅, ∅), false other way

I insertArc()I input: D = (V ,A), i , j ∈ VI output: D = (V ,A ∪ (i , j))

I insertVertex()I input: D = (V ,A), V = {0, 1, · · · , n − 1}I output: D = (V ′,A), V ′ = {0, 1, · · · , n − 1, n}



I removeArc()I input: D = (V ,A), i , j ∈ VI output: D = (V ,A–(i , j))

I removeVertex()I input: D = (V ,A), V = {0, 1, · · · , n − 1}, kI output: D = (V ′,A′), V ′ = {0, 1, · · · , n − 2}

{i ′, j ′} ∈ A′ ⇔ (∃{i , j} ∈ A) i 6= k , j 6= k,

i ′ = if (i < k) then i else i − 1,

j ′ = if (j < k) then j else j − 1



I outAdjacencyList()I input: D = (V ,A), i ∈ VI output: the list of direct successors of i

I inAdjacencyList()I input: D = (V ,A), i ∈ VI output: the list of direct predecessors of i

I listOfReachableVertices()I input: D = (V ,A), i ∈ VI output: the list of vertices reachable from i


The representation of graphs as digraphs

G = (V ,E ) =⇒ D(G ) = (V ,A){i , j} ∈ E =⇒ (i , j), (j , i) ∈ AI the topology is preserved

I the adjacency list of i in G = the out (=in) adjacency list of i in D


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The implementation of digraphs with adjaceny matrices

I the representation of digraphsI n the number of verticesI m the number of arcs (optional)I a matrix (a[i , j ]| 1 ≤ i , j ≤ n)

a[i , j ] = if (i , j) ∈ A then 1 else 0

I if the digraph is a graph, then a[i , j ] is symmetricI the out adjacency list of i ⊆ line iI the in adjacency list of i ⊆ column i



0 1 2 30 0 1 0 01 0 0 1 02 1 0 0 03 0 1 1 0



I operationsI emptyDigraph

n← 0; m← 0I insertVertex: O(n)I insertArc: O(1)I removeArc: O(1)



I removeVertex()

Procedure removeVertex(a, n, k)begin

for i ← 0 to n − 1 dofor j ← 0 to n − 1 do

if (i > k) thena[i − 1, j ]← a[i , j ]

if (j > k) thena[i , j − 1]← a[i , j ]

n← n − 1endthe execution time: O(n2)



I listOfReachableVertices()I If i = j then j is reachable from i

If i 6= j then there is a path i j if there is the arc i → j or there is k:∃i k , k j



I listOfReachableVertices()

Procedure reflTransClosure(a, n, b) // (Warshall, 1962)begin

for i ← 0 to n − 1 dofor j ← 0 to n − 1 do

b[i , j ]← a[i , j ]if (i = j) then

b[i , j ]← 1for k ← 0 to n − 1 do

for i ← 0 to n − 1 doif (b[i , k] = 1) then

for j ← 0 to n − 1 doif (b[k , j ] = 1) then

b[i , j ]← 1endthe execution time: O(n3)


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The implementation with adjacency lists

I the representation of digraphs with adjacency lists

I a vector a[0..n − 1] of linked lists (pointers)I a[i ] is the out adjacency list corresponding to i


The implementation with adjacency lists

I operationsI emptyDigraphI insertVertex: O(1)I insertArc: O(1)I removeVertex: O(n + m)I removeArc: O(m)


Content








Digraphs: systematic exploration

I it manages two setsI S = the set of already visited verticesI SB ⊆ S the subset of vertices for which there are chances to find

neighbors not visited yet

I the adjacency list of i is divided in:



I the current stepI read a vertex i from SBI extract j from the ”waiting” list of i (if it is nonempty)I if j isn’t in S , then add it to S and to SBI if the ”waiting” list of i is empty, then remove i from SB

I initiallyI S = SB = {i0}I the ”waiting” list of i = the adjacency list of i

I termination SB = ∅



Procedure exploration(a, n, i0, S)begin

for i ← 0 to n − 1 dop[i ]← a[i ]

SB ← (i0)visit(i0); S ← (i0)while (SB 6= ∅) do

i ← read(SB)if (p[i ] = NULL) then

SB ← SB − {i}else

j ← p[i ]→ varfp[i ]← p[i ]→ succif (j 6∈ S) then

SB ← SB ∪ {j}visit(j); S ← S ∪ {j}

end


Systematic exploration: complexity

TheoremAssuming that the operations over S and SB as well as visit() areachieved in O(1), the time complexity, in the worst case, of theexploration algorithm is O(n + m).


The DFS (Depth First Search) exploration

I SB is implemented as a stack

SB ← (i0)⇔SB ← emptyStack()

push(SB, i0)

i ← read(SB)⇔i ← top(SB)

SB ← SB − {i} ⇔pop(SB)

SB ← SB ∪ {j} ⇔push(SB, j)


The DFS exploration: example


The BFS (Breadth First Search) exploration

I SB is implemented as a queue

SB ← (i0)⇔SB ← emptyQueue();

insert(SB, i0)

i ← read(SB)⇔read(SB, i)

SB ← SB − {i} ⇔remove(SB)

SB ← SB ∪ {j} ⇔insert(SB, j)


The BFS exploration: example


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Finding the connected components (undirected graphs)

Function ConnectedCompDFS(D)begin

for i ← 0 to n − 1 docolor [i ]← 0

k ← 0for i ← 0 to n − 1 do

if (color [i ] = 0) thenk ← k + 1DfsRecConnectedComp(i , k)

return kend


Finding the connected components (undirected graphs)

Procedure DfsRecConnectedComp(i , k)begin

color [i ]← kfor (each vertex j in listaDeAdiac(i)) do

if (color [j ] = 0) thenDfsRecConnectedComp(j , k)

end


The strongly connected components (digraphs)


The strongly connected components: example


Finding the strongly connected components

Procedure DfsStronglyConnectedComp(D)begin

for i ← 0 to n − 1 docolor [i ]← 0parent[i ]← −1

time ← 0for i ← 0 to n − 1 do

if (color [i ] = 0) thenDfsRecStronglyConnectedComp(i)

end



Procedure DfsRecStronglyConnectedComp(i)begin

time ← time + 1color [i ]← 1for (each vertex j in adiacList(i)) do

if (color [j ] = 0) thenparent[j ]← iDfsRecStronglyConnectedComp(j)

time ← time + 1finalTime[i ]← time

end



Notation: DT = (V ,AT ), (i , j) ∈ A⇔ (j , i) ∈ AT

Procedure StronglyConnectedComp(D)begin

1. DFSStronglyConnectedComp(D)2. compute DT

3. DFSStronglyConnectedComp(DT ) but considering in the for mainloop the vertices in descending order of their final times of visitingfinalTime[i ]

4. return each tree computed at step 3 as being a distinct stronglyconnected component

end


Finding the strongly connected components: complexity

I DFSStronglyConnectedComp(D): O(n + m)

I compute DT : O(m)

I DFSStronglyConnectedComp(DT ): O(n + m)

I Total: O(n + m)


Applications

I Algorithms, path problems, computer networks (routing), genomics(alignment networks, genome assembly), multi-relational data mining,operations research (scheduling), artificial intelligence (constraintsatisfaction), etc.


Applications

The Konigsberg Bridge Problem (1736): starting from one land masses,walk over each of the seven bridges just once

The land masses: vertices, the bridges: edges

It is possible to choose a vertex, proceed along the edges and return to thechosen vertex, covering each edge once?


Applications

I Google search engine: PageRank algorithm - to determine howimportant a given web page is

I Geographic Information Systems (GIS): Google Maps, Bing Maps

I Social networks


DNA word design

I The design of DNA codes that satisfy combinatorial constraints; usein biomolecular computing to store information, or to manipulatemolecules in chemical libraries

I Find the largest set S of strings of length n over the alphabet{A,C ,G ,T} s.t.:

I GC Content Constraint: each word has 50% symbols from {C ,G}I Hamming Distance Constraint: each pair of words, w1 6= w2 differ in at

least d positions: H(w1,w2) ≥ dI Reverse Complement Hamming Distance Constraint:

H(R(w1),C (w2)) ≥ d ; R(w): the reversed of word w and C (w) thecomplement of w (C ↔ G , A↔ T )


Graph modeling

I every DNA word has an assigned vertex viI E = EHD ∪ ERC (EHD pairs of words with a HD conflict, ERC pairs of

words with a RC conflict)

I a solution: a maximum independent set

Figura: Graphs for words of size 2 and Hamming distance d = 2 (left) and d = 3(right)

AC

TCAG

TG

CA

CTGA

GT

AC

TCAG

TG

CA

CTGA

GT


Solution of 136 words for n = 8, d = 4 instance

AAACCACC ACCAGTGT ACCCAAGA ACGTAGTG ACTGACGT AGGAAGCTAGTCCTCT AGTTGGCA ATCCCGTT ATGGGCTT CAAACCTC CAAGAGACCAAGCAGT CACAGTTG CACCAATC CAGATGGT CAGGATCT CATCGTGTCATGACTG CATTCGCT CCAGTCTT CCCTGATT CCGACTTT CCTCAGTTCGAAGGTT CGACACAT CGATTTGG CGCACAAT CGCCTTTT CGCTAGTACGGTGTAT CGTAAAGG CGTGTGAT CTATGCCT CTCGTACT CTGAAGAGCTGCAAGT CTTACCGT CTTCCTAG GAAAGCGT GAACAGCT GAACGTAGGAAGGATC GACATGAG GACCTAGT GACTGTCT GAGAAGTC GAGACACTGAGTACAG GATGCAAG GATGTCCT GCAATAGG GCAGCTAT GCCTAGATGCGATCAT GCGGAATT GCTCGAAT GCTTATGG GGAAATGC GGACCATTGGATAACG GGCAACTT GGGTTGTT GGTATTCG GGTTCCAT GGTTTAGCGTAACCAG GTAGAGTG GTATCGGT GTCAGTAC GTCCAAAG GTCGATGTGTGAGATG GTGCTTCT GTTAGGCT GTTCTCTG GTTGACAC TAACACGCTAAGCTCG TACACAGC TACCGCTT TAGATCCG TAGGAAGG TAGGCGTTTAGTGTGC TATCGACG TATGTGGC TCAACGTG TCACGTCT TCAGACAGTCATGCTC TCCATGCT TCCCATTG TCCGTATC TCCTCAAG TCGAAGGATCGAGTAG TCGCAAAC TCGGTTGT TCGTACCT TCTACCAC TCTCCTGATCTCTGAG TCTGCACT TGAACCCT TGACCTAC TGAGAGGT TGATGGAGTGCAGTCA TGCGTTAG TGCTACAC TGCTCTGT TGGAGAGT TGGATGACTGGCTATG TGGGATTC TGTAGCTG TGTCTCGT TGTGACCA TGTGGAACTGTTCGTC TTAAGGGC TTACCAGG TTAGTCCC TTCAACGG TTCCTTGCTTCGCCAT TTCGGGTA TTCTGACC TTGACTCC TTGCCCTA TTGCGGATTTGTTGGG TTTCAGCC TTTGGTGG TTTTCCCG


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