Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

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1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

Why do many library sorting routines switch to insertion sort for very small subarrays?

  1. AInsertion sort uses less memory than any other sorting algorithm on large inputs
  2. BRecursive algorithms cannot handle arrays below 10 elements
  3. CInsertion sort is O(n log n) in the worst case
  4. DInsertion sort has little overhead and is fast on small or nearly sorted inputs
2.Algorithms & complexityHard

What is the time complexity of: for i = 1 to n: for (j = i; j ≤ n; j = j + i) do O(1) work?

  1. AO(n²)
  2. BO(n)
  3. CO(n √n)
  4. DO(n log n)
3.Data structuresMedium

An application needs fast lookups and also needs to list keys in sorted order and answer range queries. Which structure fits best?

  1. AA stack
  2. BA hash table
  3. CA balanced binary search tree
  4. DAn unsorted linked list
4.Data structuresHard

Deleting a node with two children from a binary search tree is usually done by:

  1. ARemoving the whole subtree
  2. BSwapping it with the root
  3. CReplacing it with its in-order successor (or predecessor) and deleting that node
  4. DRebuilding the subtree below it from scratch and reinserting every key it contained
5.Graph theoryMedium
ABCDEFEuler circuitEuler path, no circuitneither

Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.

|V| = 6|E| = 7
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
14161021163ABCDE

Find the total weight W of a minimum spanning tree of the weighted graph.

|V| = 5|E| = 7
W =

3Solutions

1.Algorithms & complexity

Answer: D · Insertion sort has little overhead and is fast on small or nearly sorted inputs

Why this is the answer

For small n the simple loop of insertion sort beats the recursion overhead of quicksort or merge sort, even though it is O(n²).

Answer

D · Insertion sort has little overhead and is fast on small or nearly sorted inputs
2.Algorithms & complexity

Answer: D · O(n log n)

Why this is the answer

The total is n / 1 + n / 2 + ... + n / n = n × Hn, and the harmonic number Hn is Θ(log n).

Answer

D · O(n log n)
3.Data structures

Answer: C · A balanced binary search tree

Why this is the answer

A balanced BST gives O(log n) lookups and keeps keys ordered, so in-order traversal and range queries are easy; a hash table scatters keys.

Answer

C · A balanced binary search tree
4.Data structures

Answer: C · Replacing it with its in-order successor (or predecessor) and deleting that node

Why this is the answer

The in-order successor has at most one child, so it can be removed easily after copying its key into the node being deleted.

Answer

C · Replacing it with its in-order successor (or predecessor) and deleting that node
5.Graph theory

Answer: Euler path, no circuit

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Degrees

deg(A) = 1, deg(B) = 4, deg(C) = 2, deg(D) = 2, deg(E) = 4, deg(F) = 1

Odd vertices

A, F → 2
Euler path, no circuit

Answer

Euler path, no circuit

The Euler path starts at one odd vertex and ends at the other.

6.Graph theory

Answer: W = 29

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Kruskal’s algorithm

BD (2): take
BE (3): take
DE (6): skip, closes a cycle
BC (10): take
CE (11): skip, closes a cycle
AB (14): take

Total weight

W = 2 + 3 + 10 + 14 = 29

Answer

W = 29

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