��2 - Chapter 2 Exercises 8 Chapter 3 Exercises...

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Unformatted text preview: Chapter 2 Exercises 8 Chapter 3 Exercises 2 ØR ‚ ´ ª ØR ‚ ´ ª ØR ‚ ´ ª = Exercises 3 ØR ‚ ´ ª xß ‚ ´ ª = Е º ·ª= a)Ø R ‚ ´ ª = b)x ß ‚ ´ ª = Е º ·ª= ØR ‚´ ª Ga B 3 % %l % %»  % X adjacent list Ø R ‚ ´ ª h uØ R ‚ ´ ª adjacent list[u] (u, v)ˆ h adjacent list[u]h vØ R ‚ ´ ª •1• ´ª= йÂ% 3.19 á t s Vì = R DFS Ø R ‚ ´ ª = ØR ‚ ´ ª MyToplogicalOrder(G=(V, E)){ for each node u in G{//O(n) inDegree[u]=0; nodes[u].type = -1; visited[u]=0; } for each edge(u,v) start from node {//O(m) inDegree[v]++; } for each node u in G{//O(n) if inDegree[u]==0 then{ add u to singleIns list; visited[u]=1; } } counter=0; while singleIns is not empty{//BFS, O(m+n) let current be the first node in the singleIns list; remove current from singleIns list; counter++; order[counter]=current; for each edge(current, v) start from current{ remove edge(current, v);//for example, remove v from adjacent list[current] inDegree[v]--; if inDegree[v]==0 then{ add v to singleIns list; visited[v]=1; } } h remove node current from graph G; //for example, set adjacent list[current]=NULL } if counter==n then{ return order; }else then{ //Law 3.19: in every DAG, there is a node v with no incoming edges //every node still in G should be in some cycles arbitrary chose a node u still in G; DFS(u);//O(n) return the subsequence of cyclePath that start and end with the same node; //O(n) } } DFS(u){ visited[u]=1; append u to cyclePath; //notation: the edges be removed is not in the adjacent list any more for each edge(u, v) still in G that start from u{ if visited[v]==1 then{ append v to cyclePath; return cyclePath; }else if visited[v]==0 then{ return DFS(v); } } remove u from cyclePath; } ¸R ‚ ´ ª = %»Â % a O(m+n)h % B3 BFS h DFS ¸ R ‚ ´ ª = ¸R ‚ ´ ª= BFS h while ¸ R ‚ ´ ª n¸ R ‚ ´ ª = for ° z … ´ ª = й % O(m)h ¸ R ‚´ ª O(n)h BFS ¸ R ‚ ´ ª = O(m+n)h DFS ¨ 3.19 ¸ R ‚ ´ ª = z Gh n-counter ¨z á ts Vì = R O(n)h Exercises 4 ¸R ‚ ´ ª °z … ´ ª = consistent ° z … ´ ª = й Â% й Â% consistent ¨z type h judgement S different ¸ R ‚ ´ ª judgements h º Â% át sVì = R judgement ¸ R ‚ ´ ª = same h % judgement h same h different »  %% a B3 % uw … ´ ª= Ð ¹Â% 0h 1 u w … ´ ª = Ð ¹  % connected tree h 0u w … ´ ª = й  % same H T ‚´ ª = 0h different H T ‚ ´ ª = (0+1)%2h H T‚´ª = connected tree H T ‚ ´ ª = falseh H T‚ ´ª ConsistentJudgment(G=(V, E)){ for each node u in G{//O(n) type[u]=-1; visited[u]=0; } treeNum=0;//number of connected trees for each node u in G{//just to chose a node to do the BFS, O(m+n) if visited[u]==0 then{ treeNum++; if treeNum>1 then{//cannot judge two nodes from different connected trees return false; } type[u]=0; visited[u]=1; add u to BFSList; while BFSList is not empty{ let current be the first node in BFSList; remove current from BFSList; for each edge(current, v) start from current{ if the type of the edge is "same" then{ tempType=type[current]; }else if the type of the edge is "different" then{ tempType=(type[current]+1)%2; } if visited[v]==0 then{ type[v]=tempType; add v to BFSList; }else if type[v]!=tempType then{//conflicts happens, not consitent return false; } } } } } return true; } HT‚ ´ª = xß‚ ´ª = Ð •º· ª = BFS O(m+n)h 3 3w … ´ ...
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