Simulating Swap Operations Without Modifying Data

In many applications, it is often required to simulate swap operations on a list of elements without actually modifying the underlying data. This can be useful in various scenarios, such as when you need to analyze the impact of different permutations on a given data structure or when you want to maintain multiple views of the data with different sorting orders.

In this blog post, we will discuss the problem of simulating swap operations without modifying the data, explore the background and applications, and provide a Python code example that demonstrates how to implement this functionality using a simple class structure. Finally, we will explain the reasons behind our implementation and conclude.

Swapping elements in a list is a fundamental operation in many algorithms, such as sorting algorithms and combinatorial search algorithms. However, there are situations where we want to simulate these swaps without actually modifying the original data. Some possible applications include:

1. Data visualization: When working with interactive visualizations, it is often necessary to display different views of the same data, based on user interactions. By simulating swaps without modifying the data, we can easily switch between different views without affecting the underlying data.
2. Algorithm analysis: Analyzing the performance of algorithms that involve swapping elements can be done more efficiently by simulating swaps without modifying the data. This allows us to observe the impact of different permutations on the algorithm's performance without the overhead of actually modifying the data structure.
3. Undo/redo functionality: In some applications, like text editors or image editing software, users may want to undo or redo certain actions. By simulating swaps without modifying the data, we can maintain a history of actions and easily revert to previous states without affecting the original data.

Suppose we have a list of elements and we want to simulate swap operations on this list without modifying the actual data. We also want to be able to retrieve the elements in their current order, reflecting the simulated swap operations.

data = ['alpha', 'beta', 'gamma', 'eta']
vec = Vector(data)

# swap 1-th and 3-th value
vec.swap(1, 3)
view = vec.view
expect_view = ['alpha', 'eta', 'gamma', 'beta']

assert all(view[i]==expect_view[i] for i in range(len(data)))

# swap 2-th and 3-th value
vec.swap(2, 3)
view = vec.view
expect_view = ['alpha', 'eta', 'beta', 'gamma']

assert all(view[i]==expect_view[i] for i in range(len(data)))

# restore data from view
restored_data = [view[i] for i in vec.addr2name]

assert all(data[i]==restored_data[i] for i in range(len(data)))

To address this problem, we propose the implementation of a class called Viewable_Mixin. This class maintains two lists, self.indices and self.inverse_indices, that store the forward and inverse mappings between the view and the data, respectively. The swap method is used to simulate swap operations on the view, while the view property returns the current state of the view.

Below is the complete Python code that demonstrates how to simulate swap operations without modifying the data using a simple class structure called Viewable_Mixin:

class Viewable_Mixin(object):
    '''Allow swap index without actually modifying the data.'''

    def __init__(self, *args, **kws):
        '''Assume data is a list'''
        super().__init__(*args, **kws)

        self.indices = list(range(len(self)))
        self.inverse_indices = list(range(len(self)))

    def swap(self, i, j):
        # swap i-th and j-th value without actually modifying data
        self.indices[i], self.indices[j] = self.indices[j], self.indices[i]

        # update inverse_indices
        self.inverse_indices[self.indices[i]] = i
        self.inverse_indices[self.indices[j]] = j

    def __getitem__(self, i):
        # return the view
        return super().__getitem__(self.indices[i])

    @property
    def view(self):
        return [self[i] for i in range(len(self))]

    @property
    def addr2name(self):
        return self.inverse_indices


class Vector(Viewable_Mixin, list):

    def __init__(self, *args, **kws):
        super().__init__(*args, **kws)


data = ['alpha', 'beta', 'gamma', 'eta']
vec = Vector(data)

# swap 1-th and 3-th value
vec.swap(1, 3)
view = vec.view
expect_view = ['alpha', 'eta', 'gamma', 'beta']

assert all(view[i] == expect_view[i] for i in range(len(data)))

# swap 2-th and 3-th value
vec.swap(2, 3)
view = vec.view
expect_view = ['alpha', 'eta', 'beta', 'gamma']

assert all(view[i] == expect_view[i] for i in range(len(data)))

# restore data from view
restored_data = [view[i] for i in vec.addr2name]

assert all(data[i]==restored_data[i] for i in range(len(data)))

Our solution is based on creating a class called Viewable_Mixin that maintains two lists: self.indices (name2addr) and self.inverse_indices (addr2name). These lists represent the forward and inverse mappings between the view and the data, respectively.

self.indices is initialized with a range of indices from 0 to the length of the data minus 1. This list represents the mapping from the view's indices to the data's indices. When we swap elements in the view, we only swap their indices in this list, without actually modifying the data.

self.inverse_indices is also initialized with a range of indices from 0 to the length of the data minus 1. This list represents the inverse mapping from the data's indices to the view's indices. It is updated whenever elements are swapped in the view, ensuring that the inverse mapping remains consistent with the forward mapping.

The swap method takes two indices, i and j, and swaps the i-th and j-th elements in the view without modifying the actual data. This is achieved by swapping the corresponding indices in self.indices and updating self.inverse_indices.

The __getitem__ method is used to return the element in the view at a given index. It does this by returning the data element at the index specified by self.indices[i].

Finally, the view and addr2name properties return the current state of the view and the inverse mapping (self.inverse_indices), respectively.

In this blog post, we have explored the problem of simulating swap operations without modifying the underlying data. We provided a Python code example that demonstrates how to achieve this using a simple class structure called Viewable_Mixin. The solution maintains two lists, self.indices and self.inverse_indices, to store the forward and inverse mappings between the view and the data. By swapping elements in the view and updating the mappings accordingly, we can efficiently simulate swaps without modifying the actual data.

This approach can be useful in various applications, such as data visualization, algorithm analysis, and undo/redo functionality, where it is necessary to maintain multiple views of the same data or analyze the impact of different permutations without affecting the underlying data.

It is possible to interprete indices and inverse_indices as two permutation matrix. To see this, one may write data and view as two column vectors and note the following equations.

Hence, indices represents the permutation matrix in the first equation, denoted by \(A\) below, and inversed_indices represents the matrix in the second equation, denoted by \(B\).

Clearly, \(AB=I\). Moreover, because \(A\) and \(B\) are othrogonal matrix, \(A=B^\intercal\) holds too.

In this point of view, indices[i] stores the unique column index j such that \(A_{ij}=1\),

\[ \sum_{j}A_{ij} \mathtt{data[j]} = \mathtt{view[i]} = \mathtt{data[indices[i]]}.\]

Similarly, inversed_indices[i] stores the unique column index j such that \(B_{ij}=1\),

\[ \sum_{j}B_{ij} \mathtt{view[j]} = \mathtt{data[i]} = \mathtt{view[inversed\_indices[i]]}.\]

For arbitary vector \(v\), we have (introduce the notation that \(\mathbb{I}[\mathtt{cond}]=1\) if and only if \(\mathtt{cond}\) is true)

Thus, we have \[i = \mathtt{inversed\_indices}[\mathtt{indices}[i]].\] This is the reflection of the matrix equation \(BA=I\).

Now look back to the equation transforming data to view

where \(\mathbb{I}(j = \mathtt{indices}[0])\) denotes a row vector with subscript \(j\). To swap the view vector without modifying the data vector, we can swap rows of \(A\) to satisfying the transforming equation. For example, if we want to swap the $i₁$-th and $i₂$-th entry of view, we can create another indices to represent the new permutation matrix:

or in a more intuitive expression

It's easy to verify that \(\mathtt{inversed\_indices}'\) is indeed the inverse mapping of \(\mathtt{indices}'\).