Show the binomial heap that results when a node with key 24 is inserted into the binomial heap shown in Figure 20.7(d). 20.1-1 20.2-3 . The time to perform BINOMIAL-HEAP-MERGE is thus O(lg n). Initially, there are at most two roots on the root list H of a given degree: because H1 and H2 were binomial heaps, they each had only one root of a given degree. (a) The situation just before line 5 of the first iteration of the while loop. We don't link x and next-x, so we simply march the pointers one position further down the list. To delete a minimum item, we remove the root and the left and right sub-trees are then merged. Chapter notes By property 2 of Lemma 20.1, the maximum depth of x is lg n, so the while loop of lines 6-10 iterates at most lg n times. Labels a, b, c, and d serve only to identify the roots involved; they do not indicate the degrees or keys of these roots. Consider a node x labeled l at depth i, and let j = k - i. In detail, the procedure works as follows. It should still take O(lg n) time. Show the binomial heap that results when a node with key 24 is inserted into the binomial heap shown in Figure 20.7(d). 3 child[z] y Write pseudocode for BINOMIAL-HEAP-MERGE. Explain why the worst-case running times of BINOMIAL-HEAP-INSERT, BINOMIAL-HEAP-MINIMUM, and BINOMIAL-HEAP-UNION are (lg n) but not (lg n). . A binomial heap H is a set of binomial trees that satisfies the following binomial-heap properties. , B0-trees. Explain why the worst-case running times of BINOMIAL-HEAP-INSERT, BINOMIAL-HEAP-MINIMUM, and BINOMIAL-HEAP-UNION are (lg n) but not (lg n). , B0-trees. At least, that's what asympotitic analysis tells us. (a) A binomial heap H. (b) The root x with minimum key is removed from the root list of H. (c) The linked list of x's children is reversed, giving another binomial heap H'. The input binomial heap H is shown in Figure 20.7(a). (e) After the link occurs, case 3 applies. This text is for readers who want to learn good programming and algorithm analysis skills simultaneously so that they can develop such programs with the maximum amount of efficiency. (e) After the link occurs, case 3 applies. We wish to find a minimum spanning tree for G: an acyclic subset T E that connects all the vertices in V and whose total weight As Figure 20.3 also shows, the roots of the binomial trees within a binomial heap are organized in a linked list, which we refer to as the root list. Node y has had its key decreased to 7, which is less than the key of y's parent z. A binomial heap is a collection of binomial trees, so this section starts by defining binomial trees and proving some key properties. 6 then min key[x] Since the root of Bk-1 has degree k-1, the root of Bk has degree k. Now by the inductive hypothesis, and as Figure 20.2(c) shows, from left to right, the children of the root of Bk-1 are roots of Bk-2, Bk-3, . Figure 20.1 Running times for operations on three implementations of mergeable heaps. 9 return y Note that the donor is emptied out: function union(b,donor) {//b and donor are binomial heaps merge the root list of the donor into the root list of b (via linked-list union) Each internal node x contains a value small[x] that is equal to the smallest key stored in any leaf in the subtree rooted at x. The following pseudocode, which can be proven correct using techniques from Section 24.1, constructs a minimum spanning tree T. It maintains a partition {Vi} of the vertices of V and, with each set Vi, a set BINOMIAL-HEAP-DELETE(H,x) (c) After another exchange and moving pointers y and z up one more level, we finally find that heap order is satisfied, so the while loop terminates. In this section, we show how to perform operations on binomial heaps in the time bounds shown in Figure 20.1. 20.2-6 BINOMIAL-HEAP-DECREASE-KEY (H,x,k) Section 20.2 shows how we can implement operations on binomial heaps in the time bounds given in Figure 20.1. Explain why the BINOMIAL-HEAP-MINIMUM procedure might not work correctly if keys can have the value . 20.2-4 20-1 2-3-4 heaps In each iteration of the while loop of lines 6-10, key[y] is checked against the key of y's parent z. In other words, the number of nodes at depth i in Bk is the number of nodes at depth i in Bk-1 plus the number of nodes at depth i - 1 in Bk-1 . If y is the root or key[y] key[z], the binomial tree is now heap-ordered. The BINOMIAL-HEAP-DELETE procedure takes O(lg n) time. 20.2-4 It is when x is the first of three that we enter case 2 in the next iteration. 20.2-4 There is no particular ordering of the keys in the leaves; that is, from left to right, the keys may be in any order. Each iteration of the while loop takes O(1) time, and there are at most 1g n1 + 1g n2 + 2 iterations because each iteration either advances the pointers one position down the root list of H or removes a root from the root list. Throughout the procedure, we maintain three pointers into the root list: Describe how to implement this algorithm using the mergeable-heap operations given in Figure 20.1. Rewrite the pseudocode to make it work correctly in such cases. The second property implies that an n-node binomial heap H consists of at most lg n + 1 binomial trees. 3 key[x] k The BINOMIAL-LINK procedure makes node y the new head of the linked list of node z's children in O(1) time. We shall only show the upper bounds; the lower bounds are left as Exercise 20.2-10. 2 sibling[y] child[z] 20.2-1 Line 10 tests for both cases 1 and 2, and lines 11-12 handle both cases. Case 1, shown in Figure 20.6(a), occurs when degree[x] degree[next-x], that is, when x is the root of a Bk-tree and next-x is the root of a Bl-tree for some l > k. Lines 11-12 handle this case. 20-1 2-3-4 heaps Extracting the node with minimum key If x is the first of two, then we enter either case 3 or case 4 in the next iteration. Otherwise, node y violates heap ordering, so its key is exchanged with the key of its parent z, along with any other satellite information. In addition, each node x contains pointers p[x] to its parent, child [x] to its leftmost child, and sibling[x] to the sibling of x immediately to its right. Give the running time of your implementation, assuming that the mergeable heaps are implemented by binomial heaps. Discuss the relationship between inserting into a binomial heap and incrementing a binary number and the relationship between uniting two binomial heaps and adding two binary numbers.
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