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

What does the Floyd–Warshall algorithm compute, and how fast?

  1. AA minimum spanning tree in O(E log V)
  2. BA topological order in O(V²)
  3. CShortest paths between all pairs of vertices in O(V³)
  4. DShortest paths from one source vertex to all others in O(V + E)
2.Algorithms & complexityHard

Shortest simple paths can be found in polynomial time. What is known about finding the longest simple path between two vertices in a general graph?

  1. AIt is always solvable in O(V + E)
  2. BIt can be solved by Dijkstra's algorithm with negated weights
  3. CIt is undecidable
  4. DIt is NP-hard
3.Data structuresMedium

How much memory does an adjacency list use for a graph with V vertices and E edges?

  1. AO(V + E)
  2. BO(V × E)
  3. CO(E²)
  4. DO(V²)
4.Data structuresHard

Which pair of traversals always determines a unique binary tree with distinct keys?

  1. ALevel-order alone
  2. BPreorder and inorder
  3. CPreorder and postorder
  4. DPreorder alone
5.Graph theoryMedium
ABCDEFGEuler 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| = 7|E| = 7
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
148361311510ABCDEF

Kruskal’s algorithm builds a minimum spanning tree. Click the edge it adds as edge number 3; skipped edges do not count.

k = 3
  1. 1edge AB
  2. 2edge AC
  3. 3edge BC
  4. 4edge BD
  5. 5edge DE
  6. 6edge DF
  7. 7edge EF
  8. 8edge BE

3Solutions

1.Algorithms & complexity

Answer: C · Shortest paths between all pairs of vertices in O(V³)

Why this is the answer

Floyd–Warshall is a dynamic programming algorithm that allows each vertex in turn as an intermediate point, using O(V³) time.

Answer

C · Shortest paths between all pairs of vertices in O(V³)
2.Algorithms & complexity

Answer: D · It is NP-hard

Why this is the answer

A longest simple path of length V − 1 is a Hamiltonian path, an NP-complete problem; on directed acyclic graphs, however, it is solvable in linear time.

Answer

D · It is NP-hard
3.Data structures

Answer: A · O(V + E)

Why this is the answer

Each vertex has a list and each edge appears once (directed) or twice (undirected), giving O(V + E).

Answer

A · O(V + E)
4.Data structures

Answer: B · Preorder and inorder

Why this is the answer

Preorder gives the root, and its position in the inorder sequence splits the left and right subtrees; preorder with postorder is ambiguous when a node has one child.

Answer

B · Preorder and inorder
5.Graph theory

Answer: neither

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) = 2, deg(C) = 2, deg(D) = 3, deg(E) = 2, deg(F) = 1, deg(G) = 3

Odd vertices

A, D, F, G → 4
neither

Answer

neither
6.Graph theory

Answer: edge BD

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

1. DF (1)
2. BC (3)
3. BD (6)
4. AC (8)
5. BE (10)

edge number 3

BD

Answer

edge BD

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