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In the course ol the construction that we present here, the branches are subdivided into three sets: I. the branchesdefinitely assignedto the tree under construction; IL the branchesfrom which the branch to be added to set I, will be selected; III. the rernaining branches'

A note on two problems in connexion with graphs

Numerische Mathematik, no. 1 (1996): 269-271

EI

We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one...更多

• The authors considera points,someor all pairs of which are connectedby a branch; the length of each branch is given.
• Construct the tree of minimum total length betweenthe n nodes.
• (A tree is a graph with one and only one path betweenevery two nodes.)
• In the course ol the construction that the authors present here, the branches are subdivided into three sets: I.

• We considera points,someor all pairs of which are connectedby a branch; the length of each branch is given
• We restrict ourselvesto the case where at least one path exists between any two nodes
• In the course ol the construction that we present here, the branches are subdivided into three sets: I. the branchesdefinitely assignedto the tree under construction; IL the branchesfrom which the branch to be added to set I, will be selected; III. the rernaining branches'
• The nodes are subdivided into two sets: A. the nodesconnectedby the branchesof set I, B. the remaining nodes
• If the branch under consideration is longer than the correspondingbranch in set II, it is rejected; it it is shorter,it replacesthe correspondingbranchin set II, and the latter is rejected
• The branchesare subdivided into three sets: L the branchesoccurring in the minimal paths ftom node P to the lodes in setA; IL the branchesfrom which the oext brancl to be placed in set I will be selected;one and only one branch of this set will lead to eachnodein set B; III. the remainingbranches

• The branchesdefinitely assignedto the tree under construction; IL the branchesfrom which the branch to be added to set I, will be selected; III.
• The nodes are subdivided into two sets: A.
• If the branch under consideration is longer than the correspondingbranch in set II, it is rejected; it it is shorter,it replacesthe correspondingbranchin set II, and the latter is rejected.
• In their solutions all the possibly ]z(z-l) - branches are first of all sorted accordingto length.
• Even if the length of the branchesis a computable function of the node coordinates,their methods demandthat data for al1 branchesare stored simultaneously.
• Find the path of rninimum total lengtl betweentwo given nodes P and Q.
• In the solution presented,the minimal paths from P to the other nodesare constructed in order of increasinglength until Q is reached.
• Iu the coue of the solution the nodes are subdivided into three sets: A.
• The nodesfor wbLichthe path o{ nrinimum lengti from P is known; nodes will be addedto this setin order of increasingminimum path length from nodeP; B.
• The branchesare subdivided into three sets: L the branchesoccurring in the minimal paths ftom node P to the lodes in setA; IL the branchesfrom which the oext brancl to be placed in set the author will be selected;one and only one branch of this set will lead to eachnodein set B; III.

• All nodesare iu set C and all branchesare in set IIL The authors transfer node P to set A and from onwardsrepeatedlyperform the following steps.
• If nodeR belongsto setB, the authors investigatewhether the use of branch z gives rise to a shorter path Jrom P to R than the known path that usesthe correspondingbranch in set II.
• If this is not so, branch r is rejected; if, use of branch z resu.ltsin a shorter connexionbetweenP and .R than hitherto obtaiaed, it replacesthe correspondingbranch in set II and the Latteris rejected.If the node.Rbelongsto setC, it is addedto setB and branch r is added to set II.
• The authors retum to stepI and repeatthe process until node Q is transferred to set A.

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