[英]Construct adjacency matrix in MATLAB
考慮布置在N×M大小的網格上的一組點。 我試圖建立鄰接矩陣,以便相鄰點連接。
例如,在帶有圖形的3x3網格中:
1-2-3
| | |
4-5-6
| | |
7-8-9
我們應該有相應的鄰接矩陣:
+---+------------------------------------------------------+
| | 1 2 3 4 5 6 7 8 9 |
+---+------------------------------------------------------+
| 1 | 0 1 0 1 0 0 0 0 0 |
| 2 | 1 0 1 0 1 0 0 0 0 |
| 3 | 0 1 0 0 0 1 0 0 0 |
| 4 | 1 0 0 0 1 0 1 0 0 |
| 5 | 0 1 0 1 0 1 0 1 0 |
| 6 | 0 0 1 0 1 0 0 0 1 |
| 7 | 0 0 0 1 0 0 0 1 0 |
| 8 | 0 0 0 0 1 0 1 0 1 |
| 9 | 0 0 0 0 0 1 0 1 0 |
+---+------------------------------------------------------+
另外,該解決方案應適用於4和8連接的相鄰點,即:
o o o o
o X o vs. o X o
o o o o
這是我到目前為止的代碼:
N = 3; M = 3;
adj = zeros(N*M);
for i=1:N
for j=1:M
k = sub2ind([N M],i,j);
if i>1
ii=i-1; jj=j;
adj(k,sub2ind([N M],ii,jj)) = 1;
end
if i<N
ii=i+1; jj=j;
adj(k,sub2ind([N M],ii,jj)) = 1;
end
if j>1
ii=i; jj=j-1;
adj(k,sub2ind([N M],ii,jj)) = 1;
end
if j<M
ii=i; jj=j+1;
adj(k,sub2ind([N M],ii,jj)) = 1;
end
end
end
如何改進以避免所有循環?
如果您注意到,正在創建的鄰接矩陣有一個獨特的模式。 具體來說,它們是對稱且帶狀的 。 您可以利用這個事實來輕松地使用diag
函數(或spdiags
函數,如果要創建稀疏矩陣)來創建矩陣。 以下是使用上面的示例矩陣作為示例的每種情況下創建鄰接矩陣的方法:
mat = [1 2 3; 4 5 6; 7 8 9]; % Sample matrix
[r, c] = size(mat); % Get the matrix size
diagVec1 = repmat([ones(c-1, 1); 0], r, 1); % Make the first diagonal vector
% (for horizontal connections)
diagVec1 = diagVec1(1:end-1); % Remove the last value
diagVec2 = ones(c*(r-1), 1); % Make the second diagonal vector
% (for vertical connections)
adj = diag(diagVec1, 1)+diag(diagVec2, c); % Add the diagonals to a zero matrix
adj = adj+adj.'; % Add the matrix to a transposed copy of
% itself to make it symmetric
您將獲得以下矩陣:
adj =
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
mat = [1 2 3; 4 5 6; 7 8 9]; % Sample matrix
[r, c] = size(mat); % Get the matrix size
diagVec1 = repmat([ones(c-1, 1); 0], r, 1); % Make the first diagonal vector
% (for horizontal connections)
diagVec1 = diagVec1(1:end-1); % Remove the last value
diagVec2 = [0; diagVec1(1:(c*(r-1)))]; % Make the second diagonal vector
% (for anti-diagonal connections)
diagVec3 = ones(c*(r-1), 1); % Make the third diagonal vector
% (for vertical connections)
diagVec4 = diagVec2(2:end-1); % Make the fourth diagonal vector
% (for diagonal connections)
adj = diag(diagVec1, 1)+... % Add the diagonals to a zero matrix
diag(diagVec2, c-1)+...
diag(diagVec3, c)+...
diag(diagVec4, c+1);
adj = adj+adj.'; % Add the matrix to a transposed copy of
% itself to make it symmetric
您將獲得以下矩陣:
adj =
0 1 0 1 1 0 0 0 0
1 0 1 1 1 1 0 0 0
0 1 0 0 1 1 0 0 0
1 1 0 0 1 0 1 1 0
1 1 1 1 0 1 1 1 1
0 1 1 0 1 0 0 1 1
0 0 0 1 1 0 0 1 0
0 0 0 1 1 1 1 0 1
0 0 0 0 1 1 0 1 0
只是為了好玩,這是一種通過計算網格上所有對點之間的距離來構造鄰接矩陣的解決方案(顯然不是最有效的方法)
N = 3; M = 3; %# grid size
CONNECTED = 8; %# 4-/8- connected points
%# which distance function
if CONNECTED == 4, distFunc = 'cityblock';
elseif CONNECTED == 8, distFunc = 'chebychev'; end
%# compute adjacency matrix
[X Y] = meshgrid(1:N,1:M);
X = X(:); Y = Y(:);
adj = squareform( pdist([X Y], distFunc) == 1 );
這是一些代碼,用於可視化鄰接矩陣和連接點圖:
%# plot adjacency matrix
subplot(121), spy(adj)
%# plot connected points on grid
[xx yy] = gplot(adj, [X Y]);
subplot(122), plot(xx, yy, 'ks-', 'MarkerFaceColor','r')
axis([0 N+1 0 M+1])
%# add labels
[X Y] = meshgrid(1:N,1:M);
X = reshape(X',[],1) + 0.1; Y = reshape(Y',[],1) + 0.1;
text(X, Y(end:-1:1), cellstr(num2str((1:N*M)')) )
我在搜索相同問題時才發現此問題。 但是,由於問題大小需要使用稀疏矩陣類型,因此所提供的解決方案都不適合我。 這是我的解決方案,適用於大規模實例:
function W = getAdjacencyMatrix(I)
[m, n] = size(I);
I_size = m*n;
% 1-off diagonal elements
V = repmat([ones(m-1,1); 0],n, 1);
V = V(1:end-1); % remove last zero
% n-off diagonal elements
U = ones(m*(n-1), 1);
% get the upper triangular part of the matrix
W = sparse(1:(I_size-1), 2:I_size, V, I_size, I_size)...
+ sparse(1:(I_size-m),(m+1):I_size, U, I_size, I_size);
% finally make W symmetric
W = W + W';
剛遇到這個問題。 我有一個很好的正常工作的m函數(鏈接: sparse_adj_matrix.m
)。
它可以處理4個連接網格(根據L1規范的半徑1),8個連接網格(根據L_infty規范的半徑1)。
它還可以支持3D(以及任意更高的三維網格)。
該函數還可以連接半徑= 1以外的節點。
這是函數的簽名:
% Construct sparse adjacency matrix (provides ii and jj indices into the
% matrix)
%
% Usage:
% [ii jj] = sparse_adj_matrix(sz, r, p)
%
% inputs:
% sz - grid size (determine the number of variables n=prod(sz), and the
% geometry/dimensionality)
% r - the radius around each point for which edges are formed
% p - in what p-norm to measure the r-ball, can be 1,2 or 'inf'
%
% outputs
% ii, jj - linear indices into adjacency matrix (for each pair (m,n)
% there is also the pair (n,m))
%
% How to construct the adjacency matrix?
% >> A = sparse(ii, jj, ones(1,numel(ii)), prod(sz), prod(sz));
%
%
% Example:
% >> [ii jj] = sparse_adj_matrix([10 20], 1, inf);
% construct indices for 200x200 adjacency matrix for 8-connect graph over a
% grid of 10x20 nodes.
% To visualize the graph:
% >> [r c]=ndgrid(1:10,1:20);
% >> A = sparse(ii, jj, 1, 200, 200);;
% >> gplot(A, [r(:) c(:)]);
您當前的代碼似乎還不錯。 您需要一種或另一種方式遍歷所有鄰居對。 如果您確實需要優化代碼,我建議:
1 <= i <= (N*M)
[iM, i+1, i+M, i-1]
請注意,要獲取所有鄰居對節點:
i % M != 0
節點的“正確”鄰居(即水平邊緣)(因為Matlab不是基於0而是基於1) i > M
“上方”鄰居(即垂直邊緣) 這將導致一個循環(但N * M迭代次數相同),不調用sub2ind(),並且循環中只有兩個if語句。
對於圖中的每個節點,在右側添加一個連接,在下方添加一個連接。 檢查您沒有超出網格范圍。 考慮以下構建鄰接矩陣的函數。
function adj = AdjMatrixLattice4( N, M )
% Size of adjacency matrix
MN = M*N;
adj = zeros(MN,MN);
% number nodes as such
% [1]---[2]-- .. --[M]
% | | |
% [M+1]-[M+2]- .. -[2*M]
% : : :
% [] [] .. [M*N]
for i=1:N
for j=1:N
A = M*(i-1)+j; %Node # for (i,j) node
if(j<N)
B = M*(i-1)+j+1; %Node # for node to the right
adj(A,B) = 1;
adj(B,A) = 1;
end
if(i<M)
B = M*i+j; %Node # for node below
adj(A,B) = 1;
adj(B,A) = 1;
end
end
end
end
上面的示例AdjMatrixLattice4(3,3)=
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
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