This algorithm finds a peak element in an array where a peak is defined as an element strictly greater than its neighbors. It uses a binary search approach to efficiently locate such an element. This is useful in scenari...

The `FindPeakElement` function implements a modified binary search to locate a peak element in an array. A peak element is guaranteed to exist because we can imagine that `nums[-1] = nums[n] = -infinity`. The algorithm iteratively narrows down the search space. If `nums[mid] < nums[mid + 1]`, it implies that a peak must exist to the right of `mid` (inclusive of `mid + 1`), so `left` is updated to `mid + 1`. Otherwise, if `nums[mid] >= nums[mid + 1]`, a peak must exist at `mid` or to its left, so `right` is updated to `mid`. The loop invariant is that a peak element is always present within the `[left, right]` range. The time complexity is O(log n) due to the binary search, and the space complexity is O(1) as it uses a constant amount of extra space. Edge cases like empty or single-element arrays are handled implicitly by the binary search logic and explicitly by an initial check.

Function FindPeakElement(array): If array is empty, throw error. Set left = 0, right = length of array - 1. While left < right: Calculate mid = left + (right - left) / 2. If array[mid] < array[mid + 1]: Set left = mid +...

This function reverses the elements of an array in-place, meaning it modifies the original array directly without allocating any additional memory for a new array. It achieves this by swapping elements from the beginning...

The `ReverseArrayInPlace` function utilizes two pointers, `left` starting at the beginning of the array and `right` starting at the end. The algorithm iteratively swaps the elements pointed to by `left` and `right` and then moves `left` one step forward and `right` one step backward. The loop continues as long as `left` is less than `right`. The loop invariant is that all elements outside the range `[left, right]` have already been swapped into their final reversed positions. The time complexity is O(n), where n is the number of elements in the array, because each element is swapped at most once. The space complexity is O(1) because the reversal is performed in-place, using only a constant amount of extra space for the temporary variable used during swapping. Edge cases such as null arrays, empty arrays, and arrays with a single element are handled by the initial conditional check, preventing any operations on invalid inputs and ensuring correctness.

Function ReverseArrayInPlace(array): If array is null or has 0 or 1 element, return. Initialize left pointer to 0. Initialize right pointer to length of array - 1. While left < right: Swap element at left with element at...

This function computes the sum of all digits in a given non-negative integer. For example, the sum of digits for 123 is 1 + 2 + 3 = 6. This is a fundamental operation used in various number theory problems, checksum calc...

The `SumOfDigits` function iteratively extracts the last digit of a number and adds it to a running sum. The modulo operator (`Mod 10`) retrieves the last digit, and integer division (`\\ 10`) effectively removes it. This process continues until the number becomes zero. The loop invariant is that `sum` holds the sum of digits processed so far, and `currentNumber` holds the remaining part of the original number. The time complexity is O(d), where d is the number of digits in the input integer, which is logarithmic with respect to the value of the integer (O(log n)). The space complexity is O(1) as it uses a fixed amount of extra memory. The function correctly handles the edge case of `n = 0` as the `While` loop condition `currentNumber > 0` will be false, and it will return the initialized `sum` of 0.

Function SumOfDigits(number): If number is negative, throw error. Initialize sum = 0. While number > 0: digit = number modulo 10. Add digit to sum. Divide number by 10 (integer division). Return sum.

This algorithm finds the minimum element in a rotated sorted array, which is an array that was originally sorted in ascending order and then rotated at some pivot point. The array might contain duplicate elements. It emp...

The `FindMin` function uses a binary search approach adapted for rotated sorted arrays, specifically handling duplicates. The core idea is to compare the middle element (`nums(mid)`) with the rightmost element (`nums(right)`). If `nums(mid) > nums(right)`, it implies that the rotation point (and thus the minimum element) lies in the right half of the array (`[mid + 1, right]`). If `nums(mid) < nums(right)`, the minimum element must be in the left half, including `mid` (`[left, mid]`). The most challenging scenario arises when `nums(mid) == nums(right)`. In this case, we cannot definitively determine which half contains the minimum. To resolve this, we safely discard the rightmost element by decrementing `right`. This is because if `nums(right)` is the minimum, `nums(mid)` is also a candidate, and we will eventually find it. If `nums(right)` is not the minimum, discarding it does not lose the true minimum. The loop invariant is that the minimum element is always contained within the `[left, right]` range. The time complexity is O(log n) on average, but degrades to O(n) in the worst case where all elements are identical due to the duplicate handling. The space complexity is O(1) as it uses a constant amount of extra space.

Function FindMin(array): If array is empty, throw error. Set left = 0, right = length of array - 1. While left < right: Calculate mid = left + (right - left) / 2. If array[mid] > array[right]: Set left = mid + 1. Else If...

This Go program defines a simplified GraphQL schema and implements a Depth First Search (DFS) algorithm to find all possible traversal paths from a specified starting field within a given type to any field matching a tar...

The `findPaths` function uses a recursive Depth First Search (DFS) approach to explore the GraphQL schema. It maintains a `visited` map to prevent infinite loops in case of cyclic schema definitions. The `dfs` helper function takes the current type and the path built so far. If the current field matches the `endField`, the path is recorded. For each field in the current type, it recursively calls `dfs` with the new path. The time complexity is roughly O(V + E) where V is the number of types and E is the number of fields in the schema, but can be worse with deep nesting and cycles. Space complexity is O(D) where D is the maximum depth of the schema, for the recursion stack and visited set.

function findPaths(schema, startType, startField, endField): initialize empty list `paths` initialize empty set `visited` function dfs(currentType, currentPath): if currentType is null: return create a unique key for cur...

This Go program implements a comprehensive GraphQL field argument validation system. It compares the arguments provided in a GraphQL field against its definition in a schema, checking for required arguments, type mismatc...

The `validateArguments` function orchestrates the validation process. It first creates a map of provided arguments for efficient lookup. It then iterates through the schema's expected arguments, checking for missing required arguments and validating the types and custom rules of provided arguments using `validateArgumentType`. Finally, it checks for any unexpected arguments not defined in the schema. The `validateArgumentType` function performs basic type checking, including handling NonNull wrappers and common scalar types. For more complex types or specific business logic, it relies on custom `Validator` functions defined in the schema. The time complexity is O(F * A) where F is the number of fields being validated and A is the average number of arguments per field, plus the complexity of custom validators. Space complexity is O(A) for storing provided arguments and errors.

function validateArguments(field, schemaField): initialize empty list `validationErrors` create map `providedArgs` from `field.Arguments` for each `argName`, `schemaArg` in `schemaField.Arguments`: get `providedArg` from...

This Go program implements a GraphQL query depth limiter. It traverses a simplified Abstract Syntax Tree (AST) representation of a GraphQL query and prunes any fields that exceed a specified maximum depth. This is a cruc...

The `limitQueryDepth` function uses recursion to traverse the query's field structure. It takes the current list of fields, the maximum allowed depth, and the current depth as parameters. If the `currentDepth` exceeds `maxDepth`, the function returns `nil`, effectively pruning that branch. Otherwise, it iterates through the fields, creating new field objects to avoid modifying the original AST. For fields with selection sets, it recursively calls `limitQueryDepth` with an incremented `currentDepth`. A field is added to the result only if its pruned selection set is not `nil` or if it's a leaf node within the depth limit. The time complexity is O(N) where N is the total number of fields in the query AST, as each field is visited once. The space complexity is O(D) where D is the maximum depth of the query, due to the recursion stack.

function limitQueryDepth(fields, maxDepth, currentDepth): if currentDepth > maxDepth: return null initialize empty list `limitedFields` for each `field` in `fields`: create a `newField` by copying `field`'s properties (N...

This Go program demonstrates a greedy approach to optimizing GraphQL field selections. It takes a list of selected fields and a map of required fields, then recursively prunes unnecessary fields. The goal is to reduce th...

The `optimizeSelections` function employs a greedy recursive strategy. For each field, it checks if the field itself is marked as required or if any of its sub-fields are required. If a field has a selection set, it recursively calls `optimizeSelections` on the sub-fields with a refined set of required sub-fields. A field is kept in the optimized selection set if it's directly required or if its optimized sub-selection set is not empty. This greedy approach prioritizes keeping any field that leads to a required piece of data, potentially leaving some fields that could be further optimized in more complex scenarios. Time complexity is roughly proportional to the number of fields and their nesting depth, as each field is visited once. Space complexity is O(D) where D is the maximum nesting depth of the selection set, due to recursion.

function optimizeSelections(fields, requiredFields): initialize empty list `optimized` for each `field` in `fields`: set `isRequired` to true if `field.Name` or `field.Alias` is in `requiredFields` if `field` has a `Sele...