ALGORITHMS & DATA STRUCTURES
EXAM QUESTIONS WITH 100%
CORRECT ANSWERS L LATEST
VERSION 2025/2026.
Greedy Algorithm - ANS an algorithm that follows problem solving heuristic of making
optimal choices at each stage. Hopefully finds the global optimum. An example would be
Kruskal's algorithm.
Divide and Conquer - ANS works by recursively breaking down a problem into two or more
sub problems until the problems become simple enough to be solved directly. An example
would be mergesort.
Recursive Algorithms - ANS solve a problem by solving smaller internal instances of a
problem -- work towards a base case.
Dynamic Programming - ANS Break down a problem into smaller and smaller subproblems.
At their lowest levels, the subproblems are solved and their answers stored in memory. These
saved answers are used again with other larger (sub)problems which may call for a
recomputation of the same information for their own answer. Reusing the stored answers
allows for optimization by combining the answers of previously solved subproblems.
Selection Sort - ANS Non-stable, in place sort. Has an N-squared order of growth, needs only
one spot of extra space. Works by searching the entire array for the smallest item, then
exchanging it with the first item in the array. Repeats this process down the entire array until it
is sorted.
1 @COPYRIGHT 2025/2026 ALLRIGHTS RESERVED.
,Insertion Sort - ANS Stable, in place sort with an order of growth which is between N and N-
squared, needs only one spot of extra space and is dependent on the order of the items. Works
by scanning over the list, then inserting the current item to the front of the list where it would
fit sequentially. All the items to the left of the list will be sorted, but may not be in their final
place as the larger items are continuously pushed back to make room for smaller items if
necessary.
ShellSort - ANS Non-stable, in place sort with an order of growth which is undetermined,
though usually given at being N-to-the 6/5. Needs only one spot of extra space. Works as an
extension of insertion sort. It gains speed by allowing exchanges of entries which are far apart,
producing partially sorted arrays which are eventually sorted quickly at the end with an
insertion sort. The idea is to rearrange the array so that every h-th entry yields a sorted
sequence. The array is h-sorted.
Quick Sort - ANS Non-stable, in place sort with an order of growth of NlogN. Needs lgN of
extra space. It has a probabilistic guarantee. Works by making use of a divide and conquer
method. The array is divided into two parts, and then the parts are sorted independently. An
arbitrary value is chosen as the partition. Afterwards, all items which are larger than this value
go to the right of it, and all items which are less than this value go to the left of it.
We arbitrarily choose a[lo] as a partitioning item. Then we scan from the left end of the array
one by one until we find an entry that is greater than a[lo]. At the same time, we are scanning
from a[lo] to the right to find an entry that is less than or equal to a[lo]. Once we find these two
values, we swap them.
3-Way Quick Sort - ANS Non-stable, in place sort with an order of growth between N and
NlogN. Needs lgN of extra space. Is probabilistic and dependent on the distribution of input
keys.
Mergesort - ANS Stable sort which is not in place. It has an order of growth of NlogN and
requires N amount of extra space. Works by dividing an array in half continuously into smaller
and smaller arrays. At the lowest level, these arrays are sorted and then merged together after
sorting in the reverse order they were divided apart in.
2 @COPYRIGHT 2025/2026 ALLRIGHTS RESERVED.
,Heap Sort - ANS Non-stable, in place sort which has an order of growth of NlogN. Requires
only one spot of extra space. Works like an improved version of selection sort. It divides its input
into a sorted and unsorted region, and iteratively shrinks the unsorted region by extracting the
smallest element and moving it into the sorted region. It will make use of a heap structure
instead of a linear time search to find the minimum.
Lower Bound on the complexity of pairwise comparisons - ANS No compare based sorting
algorithm can have fewer than ~NlogN compares
Counting Sort (Key Indexed sort) - ANS An integer sorting algorithm which counts the number
of objects that have a distinct key value, and then used arithmetic on those countes to
determine the positions of each key value in the output array.
It cannot handle large keys efficiently, and is often used as a subroutine for other sorting
algorithms such as radix sort.
Has a time complexity of N.
LSD Radix Sort - ANS Stable sort which sorts fixed length strings. Uses an axillary array, and
therefore is not in place. Goes through to the last character of a string (its least significant digit),
and takes its value. All strings given are then organized based on the value of their least
significant digit.
Following this, the algorithm proceeds to the next least significant digit, repeating the process
until it has gone through the length of the strings.
Best used for sorting things with fixed string lengths, like Social Security numbers or License
Plates.
Has a time complexity of O(n*k) where n is the number of keys and k is the average length of
those keys.
MSD Radix Sort - ANS Used to sort an array of strings based on their first character. Is done
recursively and can sort strings which are of different lengths. This algorithm will be slower than
its counterpart if used for sets of strings which all have the same length.
Has a time complexity of 2W(N+R).
3 @COPYRIGHT 2025/2026 ALLRIGHTS RESERVED.
, Unordered Linked List - ANS Data structure with non-efficently supported operations. Is
unordered. Has a worst case cost of search and insertion at N, an average case cost of insertion
at N, and an average case cost of searching at N/2.
Binary Search - ANS An ordered array of data which has efficiently supported operations. The
worst and average case of a search using this structure is lgN. The Worst case of an insertion is
N, and the average case of an insertion is N/2.
Binary Search Tree - ANS Will have a best case high of lgN. This is also its expected height. In
the worst case, it will have a height of N, and thus become similar to a linked list.
Works by inserting nodes of lesser values to the left of a node, and inserting greater values to
the right of the node, traversing down the tree until we reach a blank spot to insert.
Has a worst case cost of N to search and insert node. The average case of searching will be
1.39lgN compares
2-3 Tree - ANS Balanced tree data structure with logN complexities on searching, inserting,
and deleting in both the worst and average cases. In this data structure, every node with
children has either two children and one data element, or three children and two data
elements. Leaf nodes will contain only one or two data elements.
Red Black Tree - ANS Worst case height of 2log(n+1). The nodes are either red or black. The
root is black. If a node is red, its children MUST BE BLACK. Every path from a node to a leaf must
contain the same number of black nodes. New insertions will always be red and always left
leaning. Insertions must satisfy the conditions that red nodes have black children and that they
have the same number of black nodes in all paths. Time complexity on its operations are
O(logN).
Tries - ANS A collection of nodes, each of which can hold a key and a value- often the values
will be null. The nodes will have a value attached to the last character of the string upon
insertion, which apparently makes searching very easy.
Very useful for searching keys.
B-Trees - ANS Items are stored in leaves. The root is either a leaf, or it will have between two
and M children. All non-leaf nodes will have between M/2 and M children. All leaves will be at
4 @COPYRIGHT 2025/2026 ALLRIGHTS RESERVED.
EXAM QUESTIONS WITH 100%
CORRECT ANSWERS L LATEST
VERSION 2025/2026.
Greedy Algorithm - ANS an algorithm that follows problem solving heuristic of making
optimal choices at each stage. Hopefully finds the global optimum. An example would be
Kruskal's algorithm.
Divide and Conquer - ANS works by recursively breaking down a problem into two or more
sub problems until the problems become simple enough to be solved directly. An example
would be mergesort.
Recursive Algorithms - ANS solve a problem by solving smaller internal instances of a
problem -- work towards a base case.
Dynamic Programming - ANS Break down a problem into smaller and smaller subproblems.
At their lowest levels, the subproblems are solved and their answers stored in memory. These
saved answers are used again with other larger (sub)problems which may call for a
recomputation of the same information for their own answer. Reusing the stored answers
allows for optimization by combining the answers of previously solved subproblems.
Selection Sort - ANS Non-stable, in place sort. Has an N-squared order of growth, needs only
one spot of extra space. Works by searching the entire array for the smallest item, then
exchanging it with the first item in the array. Repeats this process down the entire array until it
is sorted.
1 @COPYRIGHT 2025/2026 ALLRIGHTS RESERVED.
,Insertion Sort - ANS Stable, in place sort with an order of growth which is between N and N-
squared, needs only one spot of extra space and is dependent on the order of the items. Works
by scanning over the list, then inserting the current item to the front of the list where it would
fit sequentially. All the items to the left of the list will be sorted, but may not be in their final
place as the larger items are continuously pushed back to make room for smaller items if
necessary.
ShellSort - ANS Non-stable, in place sort with an order of growth which is undetermined,
though usually given at being N-to-the 6/5. Needs only one spot of extra space. Works as an
extension of insertion sort. It gains speed by allowing exchanges of entries which are far apart,
producing partially sorted arrays which are eventually sorted quickly at the end with an
insertion sort. The idea is to rearrange the array so that every h-th entry yields a sorted
sequence. The array is h-sorted.
Quick Sort - ANS Non-stable, in place sort with an order of growth of NlogN. Needs lgN of
extra space. It has a probabilistic guarantee. Works by making use of a divide and conquer
method. The array is divided into two parts, and then the parts are sorted independently. An
arbitrary value is chosen as the partition. Afterwards, all items which are larger than this value
go to the right of it, and all items which are less than this value go to the left of it.
We arbitrarily choose a[lo] as a partitioning item. Then we scan from the left end of the array
one by one until we find an entry that is greater than a[lo]. At the same time, we are scanning
from a[lo] to the right to find an entry that is less than or equal to a[lo]. Once we find these two
values, we swap them.
3-Way Quick Sort - ANS Non-stable, in place sort with an order of growth between N and
NlogN. Needs lgN of extra space. Is probabilistic and dependent on the distribution of input
keys.
Mergesort - ANS Stable sort which is not in place. It has an order of growth of NlogN and
requires N amount of extra space. Works by dividing an array in half continuously into smaller
and smaller arrays. At the lowest level, these arrays are sorted and then merged together after
sorting in the reverse order they were divided apart in.
2 @COPYRIGHT 2025/2026 ALLRIGHTS RESERVED.
,Heap Sort - ANS Non-stable, in place sort which has an order of growth of NlogN. Requires
only one spot of extra space. Works like an improved version of selection sort. It divides its input
into a sorted and unsorted region, and iteratively shrinks the unsorted region by extracting the
smallest element and moving it into the sorted region. It will make use of a heap structure
instead of a linear time search to find the minimum.
Lower Bound on the complexity of pairwise comparisons - ANS No compare based sorting
algorithm can have fewer than ~NlogN compares
Counting Sort (Key Indexed sort) - ANS An integer sorting algorithm which counts the number
of objects that have a distinct key value, and then used arithmetic on those countes to
determine the positions of each key value in the output array.
It cannot handle large keys efficiently, and is often used as a subroutine for other sorting
algorithms such as radix sort.
Has a time complexity of N.
LSD Radix Sort - ANS Stable sort which sorts fixed length strings. Uses an axillary array, and
therefore is not in place. Goes through to the last character of a string (its least significant digit),
and takes its value. All strings given are then organized based on the value of their least
significant digit.
Following this, the algorithm proceeds to the next least significant digit, repeating the process
until it has gone through the length of the strings.
Best used for sorting things with fixed string lengths, like Social Security numbers or License
Plates.
Has a time complexity of O(n*k) where n is the number of keys and k is the average length of
those keys.
MSD Radix Sort - ANS Used to sort an array of strings based on their first character. Is done
recursively and can sort strings which are of different lengths. This algorithm will be slower than
its counterpart if used for sets of strings which all have the same length.
Has a time complexity of 2W(N+R).
3 @COPYRIGHT 2025/2026 ALLRIGHTS RESERVED.
, Unordered Linked List - ANS Data structure with non-efficently supported operations. Is
unordered. Has a worst case cost of search and insertion at N, an average case cost of insertion
at N, and an average case cost of searching at N/2.
Binary Search - ANS An ordered array of data which has efficiently supported operations. The
worst and average case of a search using this structure is lgN. The Worst case of an insertion is
N, and the average case of an insertion is N/2.
Binary Search Tree - ANS Will have a best case high of lgN. This is also its expected height. In
the worst case, it will have a height of N, and thus become similar to a linked list.
Works by inserting nodes of lesser values to the left of a node, and inserting greater values to
the right of the node, traversing down the tree until we reach a blank spot to insert.
Has a worst case cost of N to search and insert node. The average case of searching will be
1.39lgN compares
2-3 Tree - ANS Balanced tree data structure with logN complexities on searching, inserting,
and deleting in both the worst and average cases. In this data structure, every node with
children has either two children and one data element, or three children and two data
elements. Leaf nodes will contain only one or two data elements.
Red Black Tree - ANS Worst case height of 2log(n+1). The nodes are either red or black. The
root is black. If a node is red, its children MUST BE BLACK. Every path from a node to a leaf must
contain the same number of black nodes. New insertions will always be red and always left
leaning. Insertions must satisfy the conditions that red nodes have black children and that they
have the same number of black nodes in all paths. Time complexity on its operations are
O(logN).
Tries - ANS A collection of nodes, each of which can hold a key and a value- often the values
will be null. The nodes will have a value attached to the last character of the string upon
insertion, which apparently makes searching very easy.
Very useful for searching keys.
B-Trees - ANS Items are stored in leaves. The root is either a leaf, or it will have between two
and M children. All non-leaf nodes will have between M/2 and M children. All leaves will be at
4 @COPYRIGHT 2025/2026 ALLRIGHTS RESERVED.
