# Cormen Edition 3 Exercise 6.5 Question 9 (Page No. 166)

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Give an $O(n\lg\ k)$- time algorithm to merge $k$ sorted lists into one sorted list, where $n$ is the total number of elements in all the input lists. (Hint: Use a minheap for $k$-way merging.)

## Related questions

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The operation HEAP-DELETE$(A, i)$ deletes the item in node $i$ from heap $A$. Give an implementation of HEAP-DELETE that runs in $O(lg\ n)$ time for an $n-$element max-heap.
Each exchange operation on line $5$ of HEAP-INCREASE-KEY typically requires three assignments. Show how to use the idea of the inner loop of INSERTION-SORT to reduce the three assignments down to just one assignment.
Argue the correctness of HEAP-INCREASE-KEY using the following loop invariant: At the start of each iteration of the while loop of lines $4-6$, the subarray $A[1..A.heapsize]$ satisfies the max-heap property, except that there may be one violation$::$ $A[i]$ may be ... $A[a..heapsize]$ satisfies the max-heap property at the time HEAP-INCREASE-KEY is called.