leetcode | Data structures and algorithms for interview preparation | Learning library

Notice that you put up a bug report in GitHub and see this issue: Renaming variables didn't work, the programmer replied:

Some language features are currently not supported in notebooks, but we are making plans now to hopefully bring more of those online soon.

So please wait for this feature.

They are right. A properly implemented Union Find with path compression and union by rank has linear run time complexity as a whole, while any individual operation has an amortized constant run time complexity. The exact complexity of m operations of any type is O(m * alpha(n)) where alpha is the inverse Ackerman function. For any possible n in the physical world, the inverse Ackerman function doesn't exceed 4. Thus, we can state that individual operations are constant and algorithm as a whole linear.

The key part for path compression in your code is here:

The actual term you’re asking for is short-circuiting

Further, some operations are deemed short-circuiting operations. An intermediate operation is short-circuiting if, when presented with infinite input, it may produce a finite stream as a result. A terminal operation is short-circuiting if, when presented with infinite input, it may terminate in finite time. Having a short-circuiting operation in the pipeline is a necessary, but not sufficient, condition for the processing of an infinite stream to terminate normally in finite time.

The term “lazy” only applies to intermediate operations and means that they only perform work when being requested by the terminal operation. This is always the case, so when you don’t chain a terminal operation, no intermediate operation will ever process any element.

Finding out whether a terminal operation is short-circuiting, is rather easy. Go to the Stream API documentation and check whether the particular terminal operation’s documentation contains the sentence

This is a short-circuiting terminal operation.

allMatch has it, reduce has not.

This does not mean that such optimizations based on logic or algebra are impossible. But the responsibility lies at the JVM’s optimizer which might do the same for loops. However, this requires inlining of all involved methods to be sure that this conditions always applies and there are no side effect which must be retained. This behavioral compatibility implies that even if the processing gets optimized away, a peek(System.out::println) would keep printing all elements as if they were processed. In practice, you should not expect such optimizations, as the Stream implementation code is too complex for the optimizer.

This happens because prev's reference is being pointed by res when you do res=[prev], basically the address where the actual prev array is stored is pointed, as the prev updates, it also show changes in res.

Will std::sort always compare equal values or sometimes it can skip comparing them and therefore duplicate values will not be found?

Yes, some equal value elements will always be compared if duplicates do exist.

Let us assume the opposite: initial array of elements {e} for sorting contains a subset of elements having the same value and a valid sorting algorithm does not call comparison operator < for any pair of the elements from the subset.

Then we construct same sized array of tuples {e,k}, with the first tuple value from the initial array and arbitrary selected second tuple value k, and apply the same sorting algorithm using the lexicographic comparison operator for the tuples. The order of tuples after sorting can deviate from the order of sorted elements {e} only for same value elements, where in the case of array of tuples it will depend on second tuple value k.

Since we assumed that the sorting algorithm does not compare any pair of same value elements, then it will not compare the tuples with the same first tuple value, so the algorithm will be unable to sort them properly. This contradicts our assumptions and proves that some equal value elements (if they exist in the array) will always be compared during sorting.

We can start off by enumerating the ones. We do the by applying the function ⍸ (where, but since all are 1s, it is equivalent to 1,2,3,…) @ at the subset masked by ⊢ the bits themselves, i.e. ⍸@⊢:

It's called binomial coefficient, or "nCk" or "n Choose k".

The formula is

Here n is the size of the set, and k = 2 is the number of elements to select, so that e.g. sets {3, 6} and {6,3} taken are considered equal.

AFAIK, the standard notation in combinatorics is as shown above and spelled "n choose k", where as C(...) is non-standard requiring clarification when first introduced.

this:

When the following condition is true: