Consider the pseudocode given for DecreaseKey on a min-heap. The two loop invariants below together lead to proving that, after Decreasekey is performed, the tree satisfies the min-heap property again. Loop invariant 1: for all nodes u in H except v and its root, keyſparentfu]] < key[u). Loop invariant 2 if u has a parent, then for each child c of v, key[parent{v]] < key[c). DecreaseKey(H, v, k) Input a min-heap H containing node v Output: a min-heap in which the key of node v has been set of k 1. key[u) +k 2. while v is not the root and keyſparentſu]] > key[u] do 3. swap v and its parent a Prove maintenance of Loop invariant 1. Hint think carefully about which nodes get a new parent! Also, use Loop invariant 2.
Consider the pseudocode given for DecreaseKey on a min-heap. The two loop invariants below together lead to proving that, after Decreasekey is performed, the tree satisfies the min-heap property again. Loop invariant 1: for all nodes u in H except v and its root, keyſparentfu]] < key[u). Loop invariant 2 if u has a parent, then for each child c of v, key[parent{v]] < key[c). DecreaseKey(H, v, k) Input a min-heap H containing node v Output: a min-heap in which the key of node v has been set of k 1. key[u) +k 2. while v is not the root and keyſparentſu]] > key[u] do 3. swap v and its parent a Prove maintenance of Loop invariant 1. Hint think carefully about which nodes get a new parent! Also, use Loop invariant 2.
Computer Networking: A Top-Down Approach (7th Edition)
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Chapter1: Computer Networks And The Internet
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![2 Exercise 2
Consider the pseudocode given for Decreasekey on a min-heap. The two loop invariants below together lead
to proving that, after Decreasekey is performed, the tree satisfies the min-heap property again.
Loop invariant 1: for all nodes u in H except v and its root, key[parent{u]] < key[u).
Loop invariant 2 if u has a parent, then for each child c of U, key[parentſu]] < key[c).
Decreasekey(H, v, k)
Input a min-heap H containing node v
Output. a min-heap in which the key of node v has been set of k
1. key[v) -k
2. while v is not the root and keyſparentſu] > key[u] do
3.
swap v and its parent
a Prove maintenance of Loop invariant 1.
Hint think carefully about which nodes get a new parent! Also, use Loop invariant 2.
b
Prove termination: the resulting tree is indeed a min-heap.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe522a9f4-abc6-4fa2-bbd6-740137fb9b1e%2F3f58f553-a3bf-4baf-b9af-49f4591d2e01%2F518vna_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2 Exercise 2
Consider the pseudocode given for Decreasekey on a min-heap. The two loop invariants below together lead
to proving that, after Decreasekey is performed, the tree satisfies the min-heap property again.
Loop invariant 1: for all nodes u in H except v and its root, key[parent{u]] < key[u).
Loop invariant 2 if u has a parent, then for each child c of U, key[parentſu]] < key[c).
Decreasekey(H, v, k)
Input a min-heap H containing node v
Output. a min-heap in which the key of node v has been set of k
1. key[v) -k
2. while v is not the root and keyſparentſu] > key[u] do
3.
swap v and its parent
a Prove maintenance of Loop invariant 1.
Hint think carefully about which nodes get a new parent! Also, use Loop invariant 2.
b
Prove termination: the resulting tree is indeed a min-heap.
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