Add answer key for Debugging workshop

Covers all tutorial bugs and three C algorithm exercises:

Tutorial section (consolidated from inline expand blocks):
- Binary Search: recursive call searched upper half instead of lower half
- Quicksort: missing p1++ after swap in partition loop
- Vector: forgot to update list->__arr after realloc

Exercise 1 - Binary Tree: correct treeAdd, treeRemove (3 cases),
findMin implementations with sentinel node pattern
Exercise 2 - Binary Heap: correct heapAdd (sift up) and heapRemoveMax
(sift down) for 1-indexed max heap array
Exercise 3 - Burrows-Wheeler Transform: suffix array construction +
safe modulo pattern for C remainder operator pitfall

Also includes:
- Common bugs to watch for per exercise
- GDB and Valgrind quick reference table
- Note about C remainder vs modulo operator

PT-BR translation exists. C code is language-agnostic.
10-model QA passed (3 clean passes). Hugo build verified.

Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>

Author: Beatris Mendez Gandica

content/english/debugging/answer-key.md

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+---
+title: "Debugging - Answer Key"
+date: 2026-04-27T00:00:00-07:00
+draft: false
+weight: 15
+hidden: true
+---
+
+{{% notice note %}}
+This answer key covers all bugs from the tutorial sections and provides correct implementations for the three exercises. The tutorial bugs are shown with expand blocks in the activity pages, but this consolidates everything into one teacher reference.
+
+**Note:** The exercise implementations below are based on the algorithms described in the workshop. The actual Replit starter code may use slightly different variable names or function signatures. The logic and approach are the same.
+{{% /notice %}}
+
+## Tutorial Section Bugs
+
+### Binary Search Bug (Part 2)
+
+**The bug:** The recursive call for searching the lower half of the array incorrectly searches the upper half.
+
+**The fix:** Change the recursive call arguments so the lower half search uses `lo` and `middle - 1`:
+
+```c
+// BUGGY: searches upper half instead of lower half
+return rbin_search(arr, middle + 1, hi, element);
+
+// FIXED: searches lower half correctly
+return rbin_search(arr, lo, middle - 1, element);
+```
+
+### Quicksort Bug (Part 3)
+
+**The bug:** The `p1` pointer (tracking the "greater" element) is not incremented after swapping a smaller element into place.
+
+**The fix:** Add `p1++;` after the swap statement inside the for loop:
+
+```c
+if (arr[p2] < arr[pivot]) {
+    swap(&arr[p1], &arr[p2]);
+    p1++; // THIS LINE WAS MISSING
+}
+```
+
+### Vector Memory Bug (Part 4)
+
+**The bug:** When reallocating the array in `pushBack`, the code forgets to update `list->__arr` to point to the new memory. This causes:
+- Invalid reads/writes (accessing freed memory)
+- Double frees (freeing the same pointer twice)
+- Memory leaks (newly allocated memory is never referenced)
+
+**The fix:** After reallocating, update the pointer:
+
+```c
+int *newArr = realloc(list->__arr, newCapacity * sizeof(int));
+list->__arr = newArr; // THIS LINE WAS MISSING
+```
+
+Also add `deleteVectorInt` call at the end of main to prevent memory leaks.
+
+## Exercise 1: Binary Tree
+
+A correct binary tree implementation in C. The tree uses a sentinel node where the actual root is the sentinel's left child.
+
+### Key operations
+
+**Tree Add** - traverse to find correct position, then insert:
+
+```c
+void treeAdd(BinaryTree *tree, int value) {
+    BTNode *newNode = (BTNode *)malloc(sizeof(BTNode));
+    newNode->data = value;
+    newNode->left = NULL;
+    newNode->right = NULL;
+    newNode->parent = NULL;
+
+    // Empty tree: insert as root (left child of sentinel)
+    if (tree->__root->left == NULL) {
+        tree->__root->left = newNode;
+        newNode->parent = tree->__root;
+        return;
+    }
+
+    // Traverse to find insertion point
+    BTNode *current = tree->__root->left;
+    while (current != NULL) {
+        if (value < current->data) {
+            if (current->left == NULL) {
+                current->left = newNode;
+                newNode->parent = current;
+                return;
+            }
+            current = current->left;
+        } else {
+            if (current->right == NULL) {
+                current->right = newNode;
+                newNode->parent = current;
+                return;
+            }
+            current = current->right;
+        }
+    }
+}
+```
+
+**Find Min** - go as far left as possible:
+
+```c
+BTNode *findMin(BTNode *node) {
+    if (node == NULL) return NULL;
+    while (node->left != NULL) {
+        node = node->left;
+    }
+    return node;
+}
+```
+
+**Tree Remove** - handle 3 cases (0, 1, or 2 children):
+
+```c
+void treeRemove(BinaryTree *tree, int value) {
+    BTNode *node = treeFind(tree, value);
+    if (node == NULL) return;
+
+    // Case 1: No children (leaf)
+    if (node->left == NULL && node->right == NULL) {
+        if (node->parent->left == node)
+            node->parent->left = NULL;
+        else
+            node->parent->right = NULL;
+        free(node);
+    }
+    // Case 2: One child
+    else if (node->left == NULL || node->right == NULL) {
+        BTNode *child = (node->left != NULL) ? node->left : node->right;
+        child->parent = node->parent;
+        if (node->parent->left == node)
+            node->parent->left = child;
+        else
+            node->parent->right = child;
+        free(node);
+    }
+    // Case 3: Two children
+    else {
+        BTNode *minRight = findMin(node->right);
+        node->data = minRight->data;
+        // Remove the minRight node (it has at most 1 right child)
+        if (minRight->right != NULL) {
+            minRight->right->parent = minRight->parent;
+        }
+        if (minRight->parent->left == minRight)
+            minRight->parent->left = minRight->right;
+        else
+            minRight->parent->right = minRight->right;
+        free(minRight);
+    }
+}
+```
+
+{{% notice tip %}}
+**Common bugs to watch for:**
+- Forgetting to update parent pointers when moving nodes
+- Not handling the case where the node to remove IS the root
+- Memory leaks from not freeing removed nodes
+- The sentinel node's name `__root` is confusing — the actual root is `tree->__root->left`
+{{% /notice %}}
+
+## Exercise 2: Binary Heap (Max Heap)
+
+A correct max binary heap using a 1-indexed array.
+
+### Key index formulas (1-indexed)
+
+| Relationship | Formula |
+|-------------|---------|
+| Left child of node `i` | `2 * i` |
+| Right child of node `i` | `2 * i + 1` |
+| Parent of node `i` | `i / 2` |
+
+### Heap Add (sift up)
+
+```c
+void heapAdd(BinaryHeap *heap, int value) {
+    heap->size++;
+    heap->__arr[heap->size] = value;
+
+    // Sift up: swap with parent while larger than parent
+    int i = heap->size;
+    while (i > 1 && heap->__arr[i] > heap->__arr[i / 2]) {
+        swap(&heap->__arr[i], &heap->__arr[i / 2]);
+        i = i / 2;
+    }
+}
+```
+
+### Heap Remove Max (sift down)
+
+```c
+int heapRemoveMax(BinaryHeap *heap) {
+    int max = heap->__arr[1];
+
+    // Move last element to root
+    heap->__arr[1] = heap->__arr[heap->size];
+    heap->size--;
+
+    // Sift down: swap with larger child while smaller than children
+    int i = 1;
+    while (2 * i <= heap->size) {
+        int child = 2 * i; // left child
+        // Pick the larger child
+        if (child + 1 <= heap->size && heap->__arr[child + 1] > heap->__arr[child]) {
+            child = child + 1;
+        }
+        // Stop if heap property is satisfied
+        if (heap->__arr[i] >= heap->__arr[child]) {
+            break;
+        }
+        swap(&heap->__arr[i], &heap->__arr[child]);
+        i = child;
+    }
+
+    return max;
+}
+```
+
+{{% notice tip %}}
+**Common bugs to watch for:**
+- Off-by-one errors with 1-indexed array (root is at index 1, not 0)
+- Comparing with wrong child during sift down (must pick the LARGER child)
+- Not handling duplicates (use `>=` not just `>` when checking heap property)
+- Forgetting to decrement size after removal
+{{% /notice %}}
+
+## Exercise 3: Burrows-Wheeler Transform
+
+The BWT uses a suffix array to efficiently compute the transform.
+
+### Key insight from the workshop
+
+The last character of each rotation is found by: `original_string[(suffix_array[i] - 1 + length) % length]`
+
+{{% notice warning %}}
+**Critical C pitfall:** The `%` operator in C is the **remainder** operator, not modulo. For negative numbers, `(-1) % 6` gives `-1` in C, not `5`. Use this safe modulo pattern:
+
+```c
+// Safe modulo that works for negative numbers
+int safeMod(int a, int n) {
+    return ((a % n) + n) % n;
+}
+```
+{{% /notice %}}
+
+### Suffix Array Construction
+
+```c
+// Compare two suffixes for sorting
+int compareSuffixes(const void *a, const void *b, void *arg) {
+    char *str = (char *)arg;
+    int i = *(int *)a;
+    int j = *(int *)b;
+    return strcmp(str + i, str + j);
+}
+
+// Build suffix array: sorted indices of all suffixes
+void buildSuffixArray(char *str, int *suffixArray, int len) {
+    for (int i = 0; i < len; i++) {
+        suffixArray[i] = i;
+    }
+    // Sort suffix indices by the suffix they represent
+    // (uses qsort_r or equivalent to pass string as context)
+    qsort_r(suffixArray, len, sizeof(int), compareSuffixes, str);
+}
+```
+
+### BWT using suffix array
+
+```c
+void bwt(char *dest, char *src) {
+    int len = strlen(src) + 1; // include null terminator
+
+    int *suffixArray = malloc(len * sizeof(int));
+    buildSuffixArray(src, suffixArray, len);
+
+    // Get last character of each rotation
+    for (int i = 0; i < len; i++) {
+        int idx = ((suffixArray[i] - 1) % len + len) % len; // safe modulo
+        dest[i] = src[idx];
+    }
+
+    free(suffixArray);
+}
+```
+
+{{% notice tip %}}
+**Common bugs to watch for:**
+- Using `%` instead of safe modulo for negative numbers
+- Forgetting the null terminator adds 1 to the string length
+- Off-by-one errors in suffix array indices
+- Using `printf("%s", dest)` to print the result — this stops at the null terminator character inside the transformed string. Print character by character instead:
+
+```c
+for (int i = 0; i < STR_LEN; i++) {
+    printf("%c", dest[i]);
+}
+```
+{{% /notice %}}
+
+## Debugging Tools Quick Reference
+
+| Tool | Command | Purpose |
+|------|---------|---------|
+| Compile | `make <program>` | Build the program |
+| Run | `./<program>` | Execute the program |
+| GDB | `gdb <program>` | Start the debugger |
+| GDB: run | `run` or `r` | Run the program in GDB |
+| GDB: breakpoint | `break file.c:42` | Pause at line 42 |
+| GDB: next | `next` or `n` | Step over (skip into functions) |
+| GDB: step | `step` or `s` | Step into functions |
+| GDB: print | `print varName` | Show a variable's value |
+| GDB: continue | `continue` or `c` | Resume until next breakpoint |
+| GDB: quit | `quit` or `q` | Exit GDB |
+| Valgrind | `valgrind --tool=memcheck --leak-check=full ./<program>` | Find memory errors |